Abstract
Although the approximation of incompressible flows by finite element methods has grown quite independently of the main stream of mixed and hybrid methods, it was soon recognised that a precise analysis requires the framework of mixed methods. In many cases, one may directly apply the techniques and results of Chaps. 4 and 5. In particular, the elements used are often standard elements or simple variants of standard elements. The specificity of the Stokes problem has however led to the development of special techniques; we shall present some of them that seem particularly interesting. Throughout this study, the main point will be to make a clever choice of elements leading to the satisfaction of the inf-sup condition which is here the important one as coercivity considerations are almost always straightforward.
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Although the approximation of incompressible flows by finite element methods has grown quite independently of the main stream of mixed and hybrid methods, it was soon recognised that a precise analysis requires the framework of mixed methods. In many cases, one may directly apply the techniques and results of Chaps. 4 and 5. In particular, the elements used are often standard elements or simple variants of standard elements. The specificity of the Stokes problem has however led to the development of special techniques; we shall present some of them that seem particularly interesting. Throughout this study, the main point will be to make a clever choice of elements leading to the satisfaction of the inf-sup condition which is here the important one as coercivity considerations are almost always straightforward.
This chapter, after a quick description of the problem, will present some simple examples of elements and techniques of proof which can be used as an introduction to the subject. This will be followed by a more detailed presentation. It will not be possible to analyse in detail all the elements for which results are known; we shall try to group them by families which can be treated by similar methods. These families will be arbitrary and will overlap in many cases.
Besides this presentation of elements, we shall also consider solution techniques by penalty methods and we will develop the related problem of almost incompressible elastic materials. We shall consider the equivalence of penalty methods and mixed methods and some questions arising from it. Stabilisation techniques will also be considered.
Finally, a section will be devoted to numerical considerations and to the choice of elements.
8.1 Introduction
We have already considered, in Example 1.3.1, the Stokes problem or creeping flow problem for an incompressible fluid. We had written it as a system of variational equations: find \(\underline{u} \in V\) and p ∈ Q such that
where \(V := {(H_{0}^{1}(\Omega ))}^{n}\) and Q is the subspace of L 2(Ω) consisting of functions with zero mean value on Ω. In this formulation, \(\underline{u}\) is the velocity of the fluid and p is its pressure. A similar problem arises for the displacement of an incompressible elastic material.
Remark 8.1.1.
Although incompressibility corresponds to the case g = 0, we shall see in Remark 8.2.2 that non zero boundary conditions correspond implicitly to introducing g≠0. □
Remark 8.1.2 (Almost incompressible materials).
For a linear elastic material, following Example 1.2.2, we have to solve the variational equation
The case where λ is large, or equivalently, when the Poisson ratio ν = λ ∕ 2(λ + μ) approaches 1 ∕ 2, can be considered as an approximation of (8.1.1) by a penalty method as in Sect. 5.6.3. The limiting case is exactly (8.1.1) up to the fact that \(\underline{u}\) is a displacement instead of a velocity. Problems where λ is large are quite common and correspond to almost incompressible materials. Results of Sect. 5.5.2 can be applied and give conditions under which error estimates can be found that do not depend on λ. Problem (8.1.2) will be considered in detail in Sect. 8.12. □
It is also worth recalling that, defining
that is, \(A\underline{u} =\mathrm{ d\underline{i}v}\underline{\underline{\varepsilon }}(\underline{u})\), problems (8.1.1) and (8.1.2) are then respectively equivalent to
and
Remark 8.1.3.
The above problems are sometimes written in a simplified way. Indeed, we have, for incompressible materials,
However, this simplification of the operator is valid only if Dirichlet conditions are considered everywhere. Otherwise, the natural Neumann conditions are different and those associated with the simplified operator are unphysical. □
Remark 8.1.4.
The problems described above are, of course, physically unrealistic, as they involve body forces and homogeneous Dirichlet boundary conditions. The aim of doing so is to avoid purely technical difficulties and this implies no loss of generality. The results obtained will be valid, unless otherwise stated, for all acceptable boundary conditions. □
To approximate the Stokes problem, two approaches follow quite naturally from the preceding considerations. The first one is to use system (8.1.1) and to discretise \(\underline{u}\) and p by standard (or less standard) finite element spaces. The second one is to use formulation (8.1.2) with λ large as a penalty approximation to system (8.1.1).
It rapidly became clear that both these approaches could yield strange results. In particular, the first one often led to non convergence of the pressure (see Sect. 8.3.1) and the second one to a locking mechanism, the numerical solution being uniformly zero, or unnaturally small for λ large. For velocity-pressure approximations, empirical cures were found by Hughes and Allik [255], Hood and Taylor [249] and others. At about the same time, some elements using discontinuous pressure fields were shown to work properly [165, 200] from the mathematical point of view.
For the penalty method, the cure was found in selective or reduced integration procedures. This consisted in evaluating terms like \(\int _{\Omega }\text{div}\underline{u}\mathop{\mathrm{div}}\nolimits \underline{v}\,\mathit{dx}\) by quadrature formulae of low order. This sometimes led to good results.
It was finally stated [287], even if the result was implicit in earlier works [59], that the analysis underlying the two approaches is the same. Penalty methods are often equivalent to some mixed methods. In such cases, the penalty method works if and only if the associated mixed method works [60]. This will be developed in Sect. 8.12.
First, we must give a more precise framework to our problem.
8.2 The Stokes Problem as a Mixed Problem
8.2.1 Mixed Formulation
We shall describe in this section how the Stokes problem (8.1.1) can be analysed in the general framework of Chaps. 4 and 5. Defining \(V := {(H_{0}^{1}(\Omega ))}^{n},\) Q : = L 2(Ω) and
problem (8.1.1) can clearly be written in the form: find \(\underline{u} \in V\) and p ∈ Q such that
which is a saddle point problem in the sense of Chap. 4. Indeed, we have already seen that p is the Lagrange multiplier associated with the incompressibility constraint.
Remark 8.2.1.
It is apparent, from the definition (8.2.2) of b( ⋅, ⋅) and from the boundary conditions of the functions in V , that p, if exists, is defined up to a constant. Therefore, we change the definition of the space Q into
where two elements q 1, \(q_{2} \in {L}^{2}(\Omega )\) are identified if their difference is constant. It is not difficult to show that Q is isomorphic to the subspace of L 2(Ω) consisting of functions with zero mean value on Ω. □
With this choice, our problem reads: find \(\underline{u} \in V\) and p ∈ Q such that
Let us check the hypotheses of Theorem 4.2.2 to ensure that our problem is well-posed. Using the notation of Chap. 4, we can write
and
It can be shown (see, e.g., [362]) that
hence the operator B has a continuous lifting and the continuous inf-sup condition (4.2.26) holds. We also notice that, with our definition of the space Q, the kernel KerB t reduces to zero.
The bilinear form a( ⋅, ⋅) is coercive on V : there exists α such that
This is the well known Korn inequality (see [183, 362]), whence (4.2.12) also will follow (i.e., A is invertible on KerB).
We state the well-posedness of problem (8.2.5) in the following theorem. The proof follows from Theorem 4.2.1.
Theorem 8.2.1.
Let \(\underline{f}\) be given in \({({H}^{-1}(\Omega ))}^{n}\) and g in \(Q = L_{0}^{2}(\Omega )\) . Then, there exists a unique \((\underline{u},p) \in V \times Q\) , solution to problem (8.2.5) , which satisfies
Now, choosing an approximation V h ⊂ V and Q h ⊂ Q yields the discrete problem
The bilinear form a( ⋅, ⋅) is coercive on V ; hence, according to the theory developed in Chaps. 3 and 4, there is no problem for the existence of a solution \(\{\underline{u}_{h},p_{h}\}\) to problem (8.2.11), at least with g = 0. Indeed, we have a finite-dimensional problem where ImB is closed and the right-hand side of the second equation of (8.2.11) is zero. It should be noted, however, that we might have trouble with the uniqueness of p h and that there might be compatibility conditions on g for some approximations.
We thus try to obtain estimates of the errors \(\vert \vert \underline{u} -\underline{ u}_{h}\vert \vert _{V }\) and \(\vert \vert p - p_{h}\vert \vert _{Q}\).
First, we observe that, even for g = 0, the discrete solution \(\underline{u}_{h}\) needs not be divergence-free. Indeed, the bilinear form b( ⋅, ⋅) defines a discrete divergence operator
(it is convenient here to identify \(Q = {L}^{2}(\Omega )/\mathbb{R}\) and Q h ⊂ Q with their dual spaces). In fact, we have
and thus \(\text{div}_{h}\underline{u}_{h}\) turns out to be the L 2-projection of \(\text{div}\underline{u}\) onto Q h .
The discrete divergence operator coincides with the standard divergence operator if \(\text{div}V _{h} \subset Q_{h}\). Referring to Chap. 4, we see that obtaining error estimates requires a careful study of the properties of the operator B h = − div h and of its transpose that we denote by \(\mathop{\mathrm{g\underline{ra}d}}\nolimits _{h}\).
The first issue is to characterise the kernel \(\mathrm{Ker}B_{h}^{t} = \mathrm{Ker}(\mathop{\mathrm{g\underline{ra}d}}\nolimits _{h})\). It might happen that \(\mathrm{Ker}B_{h}^{t}\) contains non-trivial functions. In these cases, \(\mathrm{Im}B_{h} = \mathrm{Im}(\mathop{\mathrm{div}}\nolimits _{h})\) will be strictly smaller than \(Q_{h} = P_{Q_{h}}(\mathrm{Im}B)\); this may lead to pathologies. In particular, if we consider a modified problem, like the one that usually originates when dealing with Dirichlet boundary conditions, the strict inclusion \(\mathrm{Im}B_{h} \subset Q_{h}\) may even imply trouble with the existence of the solution. This situation is made clearer with the following example.
Remark 8.2.2.
Let us consider problem (8.1.4) with non-homogeneous boundary conditions, that is, let \(\underline{r}\) be such that
It is classical to reduce this case to a problem with homogeneous boundary conditions by first introducing a function \(\tilde{\underline{u}} \in {({H}^{1}(\Omega ))}^{2}\) such that \(\tilde{\underline{u}}\vert _{\Gamma } =\underline{ r}\). Setting \(\underline{u} =\underline{ u}_{0} +\tilde{\underline{ u}}\) with \(\underline{u}_{0} \in {(H_{0}^{1}(\Omega ))}^{2}\), we have to solve
with A defined in (8.1.3). We thus find a problem with a constraint \(B\underline{u}_{0} = g\) where g≠0. We have seen in Chap. 5 that the associated discrete problem may fail to have a solution because \(g_{h} = P_{Q_{h}}g\) does not necessarily belong to ImB h , whenever \(\mathrm{Ker}B_{h}^{t}\not\subset \mathrm{Ker}{B}^{t}\). Discretisations where \(\mathrm{Ker}(\mathop{\mathrm{g\underline{ra}d}}\nolimits _{h})\) is non-trivial can therefore lead to ill-posed problems in particular for some non-homogeneous boundary conditions. Examples of such conditions can be found in [340, 341]. In general, any method that relies on extra compatibility conditions is a source of trouble when applied to more complicated (non-linear, time-dependent, etc.) problems. □
For a first attempt to error estimates, we shall use Theorem 5.2.2. Since the bilinear form a( ⋅, ⋅) is coercive on V , we only have to worry about the inf-sup condition. The following proposition will be the starting point for the analysis of any finite element approximation of (8.2.5).
Proposition 8.2.1.
Let \((\underline{u},p) \in V \times Q\) be the solution of (8.2.5) and suppose the following inf-sup condition holds true
Then, there exists a unique \((\underline{u_{h}},p_{h}) \in V _{h} \times Q_{h}\) , solution to (8.2.11) , and the following estimate holds
with α given by (8.2.9) . □
Remark 8.2.3.
Actually, as it has been already observed, the existence of the discrete solution \((\underline{u_{h}},p_{h})\) (when the right-hand side in the second equation of (8.2.5) is zero) is not a consequence of the inf-sup condition (8.2.16). However, we should not forget about the possible situation presented in Remark 8.2.2. □
Of course, we shall be looking for cases where
In this case, it may be useful to summarise the estimates (8.2.17) and (8.2.18) in the following result.
Proposition 8.2.2.
With the same hypotheses as in Proposition 8.2.1 , let us suppose that (8.2.19) holds. Then, there exists C, independent of h, such that
□
Remark 8.2.4.
We shall also meet cases in which the constant k h is not bounded below by k 0. We shall then try to know precisely how it depends on h and to see whether a lower-order convergence can be achieved. When \(\mathrm{Ker}(\mathop{\mathrm{g\underline{ra}d}}\nolimits _{h})\) is non-trivial, we are interested in a weaker form of (8.2.16)
and in the dependence of k h in terms of h. From (8.2.17) and (8.2.18), one sees that the effect will be stronger on the error \(\Vert p - p_{h}\Vert _{Q}\). □
8.3 Some Examples of Failure and Empirical Cures
This section will present some classical troubles associated to the approximation of incompressible materials. We shall thus recall the difficulties associated with some ‘obvious’ approximations. We shall consider some examples of possible choices for the spaces V h and Q h , namely the \(\underline{P}_{1} - P_{1}\) element, a case of continuous pressure, and the \(\underline{P}_{1} - P_{0}\) element, a case of discontinuous pressure. These elements do not satisfy the inf-sup condition (8.2.16) and are not applicable in practice. We shall introduce some cures which will be developed and eventually justified later in this chapter.
8.3.1 Continuous Pressure: The \(\underline{P}_{1} - P_{1}\) Element
As we stated in the introduction, the development of finite element methods for incompressible problems was done, at least in the beginning, independently of the theory of mixed methods. The natural idea when attempting to solve a problem involving incompressibility would be to employ the same approximation for both velocity and pressure, in the simplest case a P 1 continuous interpolation
In the two-dimensional case, it is easy to check that if the number of triangles is large enough, then there exist non-trivial functions satisfying the discrete divergence-free condition. Thus, no locking will occur and a solution can be computed. Indeed, this method would not provide an optimal approximation of the pressures by virtue of the unbalanced approximation properties of the discrete spaces (while Q h achieves second order in L 2, V h gives only first order in H 1). On the other hand, users of such methods (you can think of using also, for instance, \((\underline{P}_{2} - P_{2})\), \((\underline{Q}_{1} - Q_{1})\), etc.) soon became aware that their results were strongly mesh dependent. In particular, the computed pressures exhibited a very strange instability. This comes from the fact that for some meshes, the kernel of the discrete gradient operator, \(\mathrm{Ker}(\mathop{\mathrm{g\underline{ra}d}}\nolimits _{h})\), is not the subspace of constant functions, as one would expect from the continuous problem, but is a larger subspace. This means that the solution obtained is determined only up to a given number of spurious pressure modes [340, 341] and that, at best, some filtering will have to be done before accurate results are available. We shall come back later on to this phenomenon also named chequerboarding in Sect. 8.10. We have already stated in Remark 8.2.2 that such spurious modes can impose non physical conditions on the data. To better understand the nature of spurious pressure modes, the reader may check the results of Fig. 8.1 in which different symbols denote points where functions in \(\mathrm{Ker}(\mathop{\mathrm{g\underline{ra}d}}\nolimits _{h})\) must have equal values for a \((\underline{P}_{1} - P_{1})\) approximation.
Spurious modes for the \(\underline{P}_{1} - P_{1}\) case
In this case, we have three spurious pressure modes. This also shows that there exists on this mesh one non-trivial discrete divergence-free function whereas a direct count would predict locking.
Remark 8.3.1 (Possible cures).
A cure for this problem was found empirically [249]: good results can be obtained using a \(\underline{P}_{2}\) approximation for velocity but a P 1 approximation for pressure. It is also clear that this does not impair the order of accuracy. This will be analysed in Sect. 8.8.
Another possibility to obtain stable elements is to add some internal degrees of freedom. The simplest case is the MINI element of [25] which we present in Sect. 8.4.2 and in a more general form in Sect. 8.5.5. □
8.3.2 Discontinuous Pressure: The \(\underline{P}_{1} - P_{0}\) Approximation
A second natural approach would be to try imposing directly the divergence-free condition. The simplest element one can imagine for the approximation of an incompressible flow would use a standard \(\underline{P}_{1}\) approximation for the velocities and a piecewise constant approximation for the pressures. With the notation of Chap. 2, this would read, again in the two-dimensional case,
As the divergence of a \(\underline{P}_{1}\) velocity field is piecewise constant, this would lead to a truly divergence-free approximation. Moreover, this would give a well-balanced O(h) approximation in estimates (8.2.17) and (8.2.18).
However, it is easy to see that such an element will not work for a general mesh. Indeed, consider a triangulation of a (simply connected) domain Ω and let us denote by
-
t the number of triangles,
-
v I the number of internal vertices,
-
v B the number of boundary vertices.
We shall thus have 2v I degrees of freedom (d.o.f.) for the space V h (since the velocities vanish on the boundary) and (t − 1) d.o.f. for Q h (because of the zero mean value of the pressures) leading to (t − 1) independent divergence-free constraints. By Euler’s relations, we have
and thus
A function \(\underline{u}_{h} \in V _{h}\) is thus over-constrained and a locking phenomenon is likely to occur: in general, the only divergence-free discrete function is \(\underline{u}_{h} \equiv 0\).
When the mesh is built under certain restrictions, it is however possible that some linear constraints become dependent: this will be the case for the cross-grid macro-element (Fig. 8.2) which will be analysed in Example 8.10.3.
The cross-grid element
As we shall see in Sect. 8.8.1, in general, obtaining truly divergence-free elements requires high degree approximations and some conditions on the mesh. We shall give, in the next section, the simplest example of a stable discontinuous pressure element, the \(\underline{P}_{2} - P_{0}\) element.
8.4 Building a B-Compatible Operator: The Simplest Stable Elements
We shall first recall here some of the results of Sect. 5.4.4 as applied to our incompressible problems. Then, we present a complete analysis of the MINI and \(\underline{P}_{2} - P_{0}\) elements and of the nonconforming \(\underline{P}_{1} - P_{0}\) elements. We shall obtain in Sect. 8.5.5 a more general proof for the MINI element.
It is not recommended to use the element \(\underline{P}_{2} - P_{0}\) because of its “unbalanced” approximation properties (O(h 2) for V h in the V -norm and only O(h) for Q h in the norm of Q), so that estimate (8.2.20) turns out to be suboptimal. However, the analysis of this element contains basic issues for getting familiar with the approximation of the Stokes problem. Moreover, the stability properties of this element will often be used as an intermediate step for the analysis of other, more efficient, elements. It must also be said that this element is not directly generalisable to the three-dimensional case.
8.4.1 Building a B-Compatible Operator
An efficient way (sometimes known as Fortin’s trick) of proving the inf-sup condition (8.2.16) consists in building a B-compatible interpolation operator Π h like in Sect. 5.4 (see [201]). We write down, here, how the hypotheses of Proposition 5.4.3 read in this particular situation.
Proposition 8.4.1.
If there exists a linear operator \(\Pi _{h} : V \rightarrow V _{h}\) such that
then the inf-sup condition (8.2.16) holds true. □
Remark 8.4.1.
As it is shown in Chap. 5, condition (8.4.1) is equivalent to \(\mathrm{Ker}(\mathop{\mathrm{g\underline{ra}d}}\nolimits _{h}) \subset \mathrm{Ker}(\mathop{\mathrm{g\underline{ra}d}}\nolimits )\). An element with this property will present no spurious pressure modes. □
In several cases, the operator Π h can be constructed in two steps as in Proposition 5.4.4. This was the case, for instance, in the proof of Proposition 8.4.3. In general, it will be enough to build two operators \(\Pi _{1},\Pi _{2} \in \mathcal{L}(V,V _{h})\) such that
where the constants c 1 and c 2 are independent of h. Then, the operator Π h satisfying (8.4.1) and (8.4.2) will be found as
In many cases, Π 1 will be the interpolation operator of [154] (cf. Chap. 2) defined in H 1(Ω).
On the contrary, the choice of Π 2 will vary from one case to the other, according to the choice of V h and Q h . However, the common feature of the various choices for Π 2 will be the following one: the operator Π 2 is constructed on each element K in order to satisfy (8.4.5). In many cases, it will be such that
We can summarise this result in the following proposition.
Proposition 8.4.2.
Let V h be such that a “Clément’s operator”: Π 1 : V → V h exists and satisfies (8.4.24) . If there exists an operator \(\Pi _{2} : V \rightarrow V _{h}\) such that (8.4.5) and (8.4.7) hold, then the operator Π h defined by (8.4.6) satisfies (8.4.1) and (8.4.2) and therefore the discrete inf-sup condition (8.2.16) holds. □
We now consider some simple examples where this construction can be used.
8.4.2 A Stable Case: The MINI Element
We now show how we can enrich the space V h of Example 8.3.1 so that, in the end, the new choice will yield a stable and convergent approximation to the Stokes problem (8.2.3). We set, as in (2.2.28),
Hence, each \(b(\underline{x})\) of B 3, on each triangle T, has the form α(T) \(\lambda _{1}(\underline{x})\) \(\lambda _{2}(\underline{x})\) \(\lambda _{3}(\underline{x})\) with α(T) constant in T. Following [25], we set
and we want to show that (8.4.9) leads to a stable and convergent approximation of the Stokes problem. For this, we are going to apply Proposition 8.4.2. We therefore have to construct an operator Π h such that
Following Proposition 8.4.2, we first take for Π 1 the operator r h of Proposition 2.2.1 and Corollary 2.2.1. We set
which, from (2.2.20), yields
In particular, (8.4.13) implies the first condition of (5.4.12)
We now define the operator \(\Pi _{2} :\, V \rightarrow {(B_{3})}^{2}\) by means of
Since \(\mathop{\mathrm{g\underline{ra}d}}\nolimits q_{h}\) is piecewise constant, (8.4.15) is easily satisfied by choosing, in each K, bubbles with the same mean value as \(\underline{v}\). It is easy to check that (under a minimum angle condition)
Indeed, for r = 1, this is the inverse inequality of Sect. 2.2.7.
From (8.4.15), it is then immediate to check that the second condition of (5.4.12) is fulfilled and, from (8.4.16) and (8.4.13), we easily have the third condition.
We can thus apply Proposition 8.4.2 and the inf-sup condition holds. Now, we apply Proposition 8.2.2 (or the more complete result of Proposition 8.2.1) and we obtain
that is, an optimal error estimate for \(\underline{u}\).
8.4.3 Another Stable Approximation: The Bi-dimensional \(\underline{P}_{2} - P_{0}\) Element
Let us now move, in the two-dimensional case, to the stable \(\underline{P}_{2} - P_{0}\) element. Precisely, we use continuous piecewise quadratic vectors for the approximation of the velocities and piecewise constants for the pressures.
The discrete divergence-free condition can then be written as
that is, as a conservation of mass on every element. This is intuitively an approximation of \(\text{div}\underline{u}_{h} = 0\), directly related to the physical meaning of this condition. It is clear from error estimates (8.2.17), (8.2.18) and standard approximation results (cf. Chap. 2) that such an approximation will lead to the loss of one order of accuracy due to the poor approximation of the pressures. However, an augmented Lagrangian technique can be used in order to recover a part of the accuracy loss (see Remark 8.4.4).
Proposition 8.4.3.
The choice
fulfils the inf-sup condition (8.2.16) .
Proof.
Before giving the rigorous proof of Proposition 8.4.3, we are going to sketch the main argument.
If we try to check the inf-sup condition by building an operator Π h satisfying (5.4.10), then, given \(\underline{u}\), we have to build \(\underline{u}_{h} = \Pi _{h}\underline{u}\) such that
Since q h is constant on every element \(K \in \mathcal{T}_{h}\), this is equivalent to
This last condition would be satisfied if \(\underline{u}_{h}\) could be built in the following way. Let us denote by M i and e i , i = 1, 2, 3, the vertices and the sides of the triangular element K (Fig. 8.3); the mid-side nodes are denoted by M ij .
The \(\underline{P}_{2}\) element
We then define
Condition (8.4.23) can be fulfilled by a correct choice of \(\underline{u}_{h}(M_{\mathit{ij}})\). Moreover, this construction can be done at element level as the choice of \(\underline{u}_{h}(M_{\mathit{ij}})\) is compatible on adjacent elements (that is, with this definition, \(\underline{u}_{h}\) turns out to be continuous).
Although this is the basic idea, some technicalities must be introduced before a real construction is obtained. Indeed, for \(\underline{u} \in {(H_{0}^{1}(\Omega ))}^{2}\), condition (8.4.22) does not make sense.
Let us then give a rigorous proof of Proposition 8.4.3. We shall rely again on Proposition 8.4.2. Denoting by \(\Pi _{1} : V \rightarrow V _{h}\) the Clément interpolant [154] described in Proposition 2.2.1, we then have
Setting r = 1 and using the triangular inequality \(\vert \vert \Pi _{1}\underline{v}\vert \vert \leq \vert \vert \underline{v} - \Pi _{1}\underline{v}\vert \vert + \vert \vert \underline{v}\vert \vert \) gives
We now modify Π 1 in a suitable way. Let us define \(\Pi _{2} : V \rightarrow V _{h}\) in the following way:
By construction, Π 2 satisfies
and a scaling argument (see Sect. 2.2.7) gives
We can now define, as in Proposition 5.4.4,
and observe that (8.4.29) and (8.4.24) imply
since
Hence, Proposition 5.4.4 applies and the proof is concluded. □
The above proof can easily be extended to more general cases. It applies to the \({(Q_{2})}^{2} - P_{0}\) quadrilateral element, provided that the usual regularity assumptions on quadrilateral meshes are made.
Remark 8.4.2 (The 2D SMALL element).
The proof will hold for elements in which only the normal component of velocity is used as a d.o.f. at the mid-side nodes [70, 198, 202]. Indeed, if only the normal component of \(\underline{u}_{h}\) is used as a degree of freedom, the \({(P_{2})}^{2} - P_{0}\) element becomes the element of Fig. 8.4 in which, on each side, the normal component of \(\underline{u}_{h}\) is quadratic, whereas the tangential component is only linear.
The 2d SMALL element
In this case, we can define \(\Pi _{2}\underline{v}\) by setting
The same remark is valid for the \({(Q_{2})}^{2} - P_{0}\) quadrilateral element. □
Remark 8.4.3.
The philosophical idea behind the \({(P_{2})}^{2} - P_{0}\) element is that we need one degree of freedom per each interface (actually, the normal component of the velocity) in order to control the jump of the pressures. This is basically the meaning of Green’s formula (8.4.21). For three-dimensional elements, however, we would need a mid-face node instead of a mid-side node in order to control the normal flux from one element to the other.
In particular, we point out that adding internal degrees of freedom to the velocity space cannot stabilise elements with piecewise constant pressures which do not satisfy the inf-sup condition. □
Remark 8.4.4.
To reduce the loss of accuracy due to the unbalanced approximation properties of the spaces V h and Q h , we can employ the augmented Lagrangian technique of Sect. 5.6.3. The discrete scheme reads: find \((\underline{u}_{h},p_{h}) \in V _{h} \times Q_{h}\) such that
Following [95], we have the following error estimate
□
8.4.4 The Nonconforming \(\underline{P}_{1} - P_{0}\) Approximation
Finally, to conclude this section about simple examples, we consider the classical (almost) stable nonconforming triangular element introduced in [165], in which mid-side nodes are used as degrees of freedom for the velocities. This generates a piecewise linear nonconforming approximation; pressures are taken constant on each element as illustrated in Fig. 8.5. It is also possible to build a three-dimensional version of this element, using mid-face nodes as degrees of freedom. We thus choose again \(Q_{h} := \mathcal{L}_{0}^{0} \cap Q\) and
\(\underline{P}_{1}^{NC} - P_{0}\) element
We remark that this method is attractive for several reasons. In particular, the restriction to an element K of the solution \(\underline{u}_{h} \in V _{h}\) is exactly divergence-free, since divV h ⊂ Q h .
As we have a nonconforming element, we must define discrete bilinear forms,
and consider the problem
Remark 8.4.5 (Problem with coercivity).
It must also be recalled that coercivity is a problem for the \(\underline{P}_{1}^{NC} - P_{0}\) element. The trouble is that the bilinear form (8.2.1) is not coercive on the nonconforming space V h and we do not have the discrete version of Korn’s inequality. This issue has been deeply investigated and clearly illustrated in [16]. It is important to note that (8.4.37) is not the same as in (8.2.1). As we stated earlier, the modified problem is valid only for Dirichlet boundary conditions. Even for the Stokes problem, the inf-sup condition is not always the only relevant one. □
Nevertheless, let us see how we can show the inf-sup condition. We may now construct \(\Pi _{h} :\, V \rightarrow V _{h}\) by
and, again, it is easy to see that
(where as usual \(\vert v_{h}\vert _{1,h}^{2} =\sum _{K}\vert v_{h}\vert _{1,K}^{2}\)) and
which implies, by Proposition 5.4.2,
On the other hand, we also have
We may now apply Proposition 5.5.6 and get
where
so that, in the end, we have the optimal estimate
The present element has been generalised to second order in [209]. In this case, there is no problem with coercivity.
Remark 8.4.6.
The generalisation of nonconforming finite elements to quadrilaterals is not straightforward. In particular, approximation properties of the involved spaces are not obvious. More details can be found in [330]. □
8.5 Other Techniques for Checking the inf-sup Condition
Having presented a few simple examples, we now consider, in a more systematic way, standard techniques for the proof of the inf-sup stability condition (8.2.16) that can be applied to a large class of elements. For ease of presentation, in this section, we develop the theory and postpone the examples to Sects. 8.6 and 8.7, for two- and three-dimensional schemes, respectively. However, after the description of each technique, we list some schemes to which that technique can be applied.
8.5.1 Projection onto Constants
Following [116], we now consider a modified inf-sup condition
where \(\bar{q}_{h}\) is the L 2-projection of q h onto \(\mathcal{L}_{0}^{0}\) (that is, piecewise constant functions).
Proposition 8.5.1.
Let us suppose that the modified inf-sup condition (8.5.1) holds with k 0 independent of h. Assume moreover that V h is such that, for any \(q_{h} \in \mathcal{L}_{0}^{0} \cap Q\),
with γ 0 independent of h. Then, the inf-sup condition (8.2.16) holds true.
Proof.
For any q h ∈ Q h , one has
which implies
Putting together (8.5.1) and (8.5.4) proves the proposition. □
Remark 8.5.1.
In the case of continuous pressures schemes, hypothesis (8.5.2) can be replaced with the following approximation assumption: for any \(\underline{v} \in V\) there exists \(\underline{{v}}^{I} \in V _{h}\) such that
The details of the proof can be found in [116] when the mesh is quasi-uniform. The quasi-uniformity assumption is actually not needed, as it can be shown with an argument similar to the one which will be presented in the next subsection (see, in particular, Remark 8.5.2). □
Example 8.5.1.
The technique presented in this section will be used, for instance, for the stability proof of the generalised two-dimensional Hood–Taylor element (see Sect. 8.8.2 and Theorem 8.8.1). □
8.5.2 Verfürth’s Trick
Verfürth’s trick [375], already presented in Sect. 5.4.5, applies to continuous pressure approximations and is essentially based on two steps. The first step is quite general and can be summarised in the following Lemma.
Lemma 8.5.1.
Let Ω be a bounded domain in \({\mathbb{R}}^{n}\) with Lipschitz continuous boundary. Let \(V _{h} \subset {(H_{0}^{1}(\Omega ))}^{n} =: V\) and \(Q_{h} \subset {H}^{1}(\Omega ) \cap Q\) be closed subspaces. Assume that there exists a linear operator \(\Pi _{h}^{0}\) from V into V h and a constant c (independent of h) such that
Then, there exist two positive constants c 1 and c 2 such that, for every q h ∈ Q h,
Proof.
Given q h ∈ Q h , let \(\bar{\underline{v}} \in V\) be such that
where β is the continuous inf-sup constant. Then,
□
Remark 8.5.2.
Indeed, via a scaling argument, it can be shown that the last term in the right-hand side of equation (8.5.7) is equivalent to \(\vert \vert q_{h} -\bar{ q}_{h}\vert \vert _{0}\), where \(\bar{q}_{h}\) denotes, as in the previous subsection, the L 2-projection onto the piecewise constants. □
We are now in the position of stating the main result of this subsection. Note that Verfürth’s trick consists in proving a kind of inf-sup condition where the zero norm of q h is substituted by \(h\vert q_{h}\vert _{1}\).
Proposition 8.5.2.
Suppose that the hypotheses of Lemma 8.5.1 hold true. Assume, moreover, that there exists a constant c 3 such that, for every q h ∈ Q h,
Then, the standard inf-sup condition (8.2.16) holds true.
Proof.
Let us multiply (8.5.7) by c 3 and (8.5.10) by c 2 and sum up the two equations. We have
that is, the inf-sup condition (8.2.16). □
Example 8.5.2.
The Verfürth trick has been designed for the stability analysis of the Hood–Taylor method. It will be used for this purpose in Sect. 8.8.2 (see Theorem 8.8.1). □
8.5.3 Space and Domain Decomposition Techniques
Sometimes, the spaces V h and Q h decompose into the sum (direct or not) of subspaces for which it might be easier to prove an inf-sup condition. This is the case, for instance, when a domain decomposition technique is employed. Some of the results we are going to present can be viewed as a particular case of the macro-element technique which will be introduced in Sect. 8.5.4.
The next result has been presented and proved in [223].
Proposition 8.5.3.
Suppose Ω can be decomposed as the union of disjoint sub-domains with Lipschitz continuous boundaries
We make use of the following notation:
Suppose, moreover, that the spaces V 0,r and Q 0,r satisfy the following inf-sup condition
with k r independent of h (r = 1,…,R) and that the following inf-sup condition between V h and K holds true
with k K independent of h. Then, the spaces V h and Q h satisfy the inf-sup condition (8.2.16) . □
Sometimes, it is not possible (or it is not the best choice) to partition Ω into disjoint sub-domains. Let us describe the case of two overlapping sub-domains. The following proposition can be checked by a direct computation.
Proposition 8.5.4.
Let Ω be the union of two sub-domains Ω 1 and Ω 2 with Lipschitz continuous boundaries. With the notation of the previous proposition, suppose that the spaces V 0,r and Q 0,r satisfy the inf-sup conditions
for r = 1,2. Then, the spaces V h and Q h satisfy the condition
where, as in Sect. 8.5.1 , we have denoted by \(\overline{q}_{h}\) the L 2 projection of q h onto the space \(\mathcal{L}_{0}^{0}\) . □
Another useful technique for proving the inf-sup condition can be found in [328]. This result is quite general; in particular, the decomposition of the spaces V h and Q h does not rely on a decomposition of the domain Ω. In [328], the following proposition is stated for a two-subspaces decomposition, but it obviously extends to more general situations.
Proposition 8.5.5.
Let Q 1 and Q 2 be subspaces of Q h such that
If V 1 , V 2 are subspaces of V h and α 1 , α 2 positive constants such that
and β 1 , β 2 are non-negative constants such that
with
then the inf-sup condition (8.2.16) holds true with k 0 depending only on α i , β i , i = 1,2. □
Remark 8.5.3.
Condition (8.5.21) is trivially true, for instance, when β 1 β 2 = 0 and α 1 α 2 > 0. □
Example 8.5.3.
Most of the techniques presented in this section can be seen as a particular case of the macro-element technique (see Sect. 8.5.4). Proposition 8.5.4 will be used in Theorem 8.8.1 for the stability proof of the Hood–Taylor scheme. □
8.5.4 Macro-element Technique
In this section we present a technique introduced by Stenberg (see [351–355]) which, under suitable hypotheses, reduces the matter of checking the inf-sup condition (8.2.16) to an algebraic problem. We also refer to [98] for related results in a somewhat different setting.
The present technique is based on a decomposition of the triangulation \(\mathcal{T}_{h}\) into disjoint macro-elements, where we refer to a macro-element as an open polygon (resp., polyhedron in \({\mathbb{R}}^{3}\)) which is the union of adjacent elements.
Let us introduce some notation.
A macro-elementM is said to be equivalent to a reference macro-element \(\hat{M}\) if there exists a mapping \(F_{M} :\hat{ M} \rightarrow M\) such that
-
1.
F M is continuous and invertible;
-
2.
\(F_{M}(\hat{M}) = M\);
-
3.
If \(\hat{M} = \cup \hat{K}_{j}\), where K j , j = 1, … m, are the elements defining \(\hat{M}\), then \(K_{j} = F_{M}(\hat{K}_{j})\), j = 1, … m, are the elements of M;
-
4.
\(F_{M}\vert _{\hat{K}_{j}} = F_{K_{j}} \circ F_{\hat{K}_{j}}^{-1}\), j = 1, … m, where F K denotes the affine mapping from the reference element to a generic element K.
We denote by \(\mathcal{E}_{\hat{M}}\) the equivalence class of \(\hat{M}\). We now introduce the discrete spaces associated with V h and Q h on the generic macro-element M (N is the dimension of Ω):
We finally introduce a space which corresponds to the kernel of \(B_{h}^{t}\) on the macro-element M:
The macro-elements condition reads
that is, the analogous (at a macro-element level) of the necessary condition for the discrete Stokes problem to be well-posed that the kernel of \(B_{h}^{t}\) reduces to the zero function.
Proposition 8.5.6.
Suppose that each triangulation \(\mathcal{T}_{h}\) can be decomposed into disjoint macro-elements belonging to a fixed number (independent of h) of equivalence classes \(\mathcal{E}_{\hat{M}_{i}}\) , i = 1,…n. Suppose, moreover, that the pair \(V _{h} -\mathcal{L}_{0}^{0}/\mathbb{R}\) is a stable Stokes element, that is,
with β independent of h. Then, the macro-element condition (8.5.24) (for every \(M \in \mathcal{E}_{\hat{M}_{i}}\) , i = 1,…n) implies the inf-sup condition (8.2.16) .
Proof.
We do not give the technical details of the proof, for which we refer to [351]. The basic arguments of the proof are sketched in Remark 8.5.4. □
Remark 8.5.4.
The macro-element condition (8.5.24) is strictly related to the patch test used by engineers (cf., e.g., [388]). However, the count of the degrees of freedom is clearly insufficient by itself. Hence, let us point out how the hypotheses of Proposition 8.5.6 are important.
Hypothesis (8.5.24) (the macro-element condition) implies, via a compactness argument, that a discrete inf-sup condition holds true between the spaces V 0, M and Q 0, M . The finite number of equivalent macro-elements classes is sufficient to conclude that the corresponding inf-sup constants are uniformly bounded below by a positive number.
Then, we are basically in the situation of the domain decomposition technique of Sect. 8.5.3. We now use hypothesis (8.5.25) to control the constant functions on each macro-element and to conclude the proof. □
Remark 8.5.5.
Hypothesis (8.5.25) is satisfied in the two-dimensional case whenever V h contains piecewise quadratic functions (see Sect. 8.3). In the three-dimensional case, things are not so easy: to control the constants, we need extra degrees of freedom on the faces, as observed in Remark 8.4.3. For this reason, let us state the following proposition which can be proved with the technique of Sect. 8.5.1 (see Remark 8.5.1) and which applies to the case of continuous pressures approximations. □
Proposition 8.5.7.
Let us make the same assumptions as in Proposition 8.5.6 with (8.5.25) replaced by the condition of Remark 8.5.1 (see (8.5.5) ). Then, provided \(Q_{h} \subset {C}^{0}(\Omega )\) , the inf-sup condition (8.2.16) holds true. □
Remark 8.5.6.
The hypothesis that the macro-element partition of \(\mathcal{T}_{h}\) is disjoint can be weakened, in the spirit of Proposition 8.5.4, by requiring that each element K of \(\mathcal{T}_{h}\) belongs at most to a finite number N of macro-elements with N independent of h. □
Example 8.5.4.
The macro-element technique can be used in order to prove the stability of several schemes. Among those, we recall the \(\underline{Q}_{2} - P_{1}\) element (see Sect. 8.6.3) and the three-dimensional generalised Hood-Taylor scheme (see Theorem 8.8.2). □
8.5.5 Making Use of the Internal Degrees of Freedom
This subsection presents a general framework providing a general tool for the analysis of finite element approximations to problems of incompressible materials.
The basic idea has been used several times on particular cases, starting from [165] for discontinuous pressures and from [24] and [25] for continuous pressures. We are going to present it in its final general form given by Brezzi and Pitkiäranta [131]. It consists essentially in stabilising an element by adding suitable bubble functions to the velocity field.
In order to do that, following the notation of Remark 2.2.4, we first associate to every finite element discretisation Q h ⊂ Q the space
where b K is a bubble function defined in K. In particular, we can take b K = b 3, K as the standard cubic bubble if K is a triangle, or a bi-quadratic bubble if K is a square or other obvious generalisations in 3D. In other words, the restriction of a \(\underline{\beta }\in B(b_{K}\mathop{ \mathrm{g\underline{ra}d}}\nolimits Q_{h})\) to an element K is the product of the bubble functions b K times the gradient of a function of \(Q_{h}\vert _{K}\).
Remark 8.5.7.
In (8.5.26), we take the gradient on K so that \(B(b_{K}\mathop{ \mathrm{g\underline{ra}d}}\nolimits Q_{h})\) is well defined even if Q h is a space of discontinuous pressures. □
Remark 8.5.8.
Notice that the space \(B(b_{k}\mathop{ \mathrm{g\underline{ra}d}}\nolimits Q_{h})\) is not defined through a basic space \(\hat{B}\) on the reference element. This could be easily done in the case of affine elements, for all the reasonable choices of Q h . However, this is clearly unnecessary: if we know how to compute q h on K, we also know how to compute \(\mathop{\mathrm{g\underline{ra}d}}\nolimits q_{h}\) and there is no need for a reference element. □
We can now prove our basic results, concerning the two cases of continuous or discontinuous pressures.
Proposition 8.5.8 (Stability of continuous pressure elements).
Assume that there exists an operator \(\Pi _{1} \in \mathcal{L}(V,V _{h})\) satisfying the property of the Clément interpolant (8.4.24) . If \(Q_{h} \subset {C}^{0}(\Omega )\) and V h contains the space \(B(b_{k}\mathop{ \mathrm{g\underline{ra}d}}\nolimits Q_{h})\) , then the pair \((V _{h},Q_{h})\) is a stable element, in the sense that it satisfies the inf-sup condition (8.2.16) .
Proof.
We shall build a B-compatible operator, like in Proposition 8.4.2. We only need to construct the operator Π 2. We define \(\Pi _{2} : V \rightarrow B(b_{k}\mathop{ \mathrm{g\underline{ra}d}}\nolimits Q_{h})\), on each element, by requiring
Problem (8.5.27) has obviously a unique solution and Π 2 satisfies (8.4.5). Finally, (8.4.7) follows by a scaling argument. Hence, Proposition 8.4.2 gives the desired result. □
Corollary 8.5.1.
Assume that Q h ⊂ Q is a space of continuous piecewise smooth functions. If V h contains \({(\mathcal{L}_{1}^{1})}^{2} \oplus B(b_{K}\mathop{ \mathrm{g\underline{ra}d}}\nolimits Q_{h})\) , then the pair \((V _{h},Q_{h})\) satisfies the inf-sup condition (8.2.16) .
Proof.
Since V h contains piecewise linear functions, there exists a Clément interpolant Π 1 satisfying (8.4.24). Hence, we can apply Proposition (8.5.8). □
We now consider the case of discontinuous pressure elements.
Proposition 8.5.9 (Stability of discontinuous pressure elements).
Assume that there exists an operator \(\tilde{\Pi }_{1} \in \mathcal{L}(V,V _{h})\) satisfying
If V h contains \(B(b_{K}\mathop{ \mathrm{g\underline{ra}d}}\nolimits Q_{h})\) , then the pair \((V _{h},Q_{h})\) is a stable element, in the sense that it satisfies the inf-sup condition (8.2.16) .
Proof.
We are going to use Proposition 8.5.8. We take \(\tilde{\Pi }_{1}\) as operator Π 1. We are not defining Π 2 on the whole V , but only in the subspace
This will be enough, since we need to apply Π 2 to the difference \(\underline{v} -\tilde{ \Pi }_{1}\underline{v}\) which is in V 0 by (8.5.28).
For every \(\underline{v} \in {V }^{0}\), we define \(\Pi _{2}\underline{v} \in B(b_{K}\mathop{ \mathrm{g\underline{ra}d}}\nolimits Q_{h})\) by requiring that, in each element K,
Note that (8.5.30) is uniquely solvable, if \(\underline{v} \in {V }^{0}\), since the divergence of a bubble function has always zero mean value (hence, the number of non-trivial equations is equal to \(\dim (Q_{h}\vert _{K}) - 1\), which is equal to the number of unknowns; the non-singularity then follows easily). It is obvious that Π 2, as given by (8.5.30), will satisfy (8.4.5) for all \(\underline{v} \in {V }^{0}\). We have to check that
which actually follows by a scaling argument making use of the following bound
□
Corollary 8.5.2 (Two-dimensional case).
Assume that Q h ⊂ Q is a space of piecewise smooth functions. If V h contains \({(\mathcal{L}_{2}^{1})}^{2} \oplus B(b_{K}\mathop{ \mathrm{g\underline{ra}d}}\nolimits Q_{h})\) , then the pair (V h ,Q h ) satisfies the inf-sup condition (8.2.16) .
Proof.
The stability of the \({(P_{2})}^{2} - P_{0}\) element (see Sect. 8.3) implies the existence of \(\tilde{\Pi }_{1}\) as in Proposition 8.5.9. □
Propositions 8.5.8 and 8.5.9 are worth a few comments. They show that almost any element can be stabilised by using bubble functions. For continuous pressure elements, this procedure is mainly useful in the case of affine elements. For discontinuous pressure elements, it is possible to stabilise elements which are already stable for a piecewise constant pressure field. Examples of such a procedure can be found in [199]. Stability with respect to piecewise constant pressure implies that at least one degree of freedom on each side or face of the element is linked to the normal component of velocity (see [202] and Remark 8.4.3).
Example 8.5.5.
Internal degrees of freedom can be used in the stability analysis of several methods. For instance, we use it for the analysis of the MINI element (see Sects. 8.6.1 and 8.7.1) in the case of continuous pressures and of the Crouzeix-Raviart element (see Proposition 8.6.2 and Sect. 8.7.2) in the case of discontinuous pressures. □
8.6 Two-Dimensional Stable Elements
In this section, we shall make use of the techniques presented in Sect. 8.5 to prove the stability for some of the most popular two-dimensional Stokes elements. The degrees of freedom corresponding to some of those are collected in Fig. 8.6.
Some stable two-dimensional Stokes elements: (a) the MINI element, (b) the Crouzeix–Raviart element, (c) the \(\underline{P}_{1}^{NC} - P_{0}\) element, (d) the \(\underline{Q}_{2} - P_{1}\) element
We start with triangular elements and then we present schemes based on quadrilaterals.
The Hood–Taylor element (two- and three-dimensional) and its generalisation will be presented in Sect. 8.8. Figure 8.6 presents the most simple cases of the elements that we shall discuss.
8.6.1 Continuous Pressure Elements
We have already presented in Sect. 8.4.2 the MINI element. This element, which is probably the simplest one for the approximation of the Stokes equation, has been introduced in [25]. Using the results of Sect. 8.5.5, in particular Corollary 8.5.1, we easily deduce the following result.
Proposition 8.6.1.
The following pair of spaces are stable for any k ≥ 1
□
For k = 1, (8.6.1) is the MINI element while (8.6.2) defines a variant of the Hood-Taylor element, which we shall consider in Sect. 8.8, enriched by bubbles. This produces an element with a slightly better k h in the inf-sup condition.
8.6.2 Discontinuous Pressure Elements
We have already considered in Sect. 8.4.3 the element \(\underline{P}_{2} - P_{0}\). Using a P 0 pressure ensures an element-wise conservation of mass which is an advantage in some situations. We now rely on Proposition 8.5.9 and more precisely on Corollary 8.5.2.
Example 8.6.1 (The Crouzeix-Raviart element).
This element, presented in [165], is an enrichment to the \(\underline{P}_{2} - P_{0}\) scheme which provides well-balanced approximation properties. Given a mesh of triangles, the approximating spaces are
The proof of the stability for this element is a direct consequence of Proposition 8.5.8. □
The Crouzeix-Raviart element is the simplest one of a general family. Indeed, the construction of the Crouzeix-Raviart element relies on the fact that, as we have seen in Proposition 8.5.9 and Corollary 8.5.2, adding enough bubbles (that is, internal degrees of freedom) to an element which is stable for pressures in \(\mathcal{L}_{0}^{0}\) can make it stable for pressures in \(\mathcal{L}_{k}^{0}\). We can thus state the following proposition.
Proposition 8.6.2.
For k ≥ 2, let
Then, the couple V h × Q h is stable. □
Remark 8.6.1 (A nonconforming version).
It can easily be checked [209] that one obtains a stable element of second order accuracy by replacing the standard bubble by the nonconforming bubble of (2.2.39), that is, taking
Besides some nice continuity properties of the stress at mid-side nodes, the fact that second order polynomials are employed simplifies things for the numerical integration when building the discrete problem. □
Remark 8.6.2.
Instead of enriching V h , stability can be obtained by taking a smaller Q h , in general to the expense of accuracy. It can easily be checked that if we take, for k ≥ 2, \(V _{h} = {(\mathcal{L}_{k}^{1})}^{2}\) and \(Q_{h} = \mathcal{L}_{k-2}^{0}\), we have enough internal degrees of freedom in V h to ensure stability. This is not true in the three-dimensional case where one would need \(Q_{h} = \mathcal{L}_{k-3}^{0}\) and, evidently, k ≥ 3. This choice of elements would have a severe impact on accuracy. □
8.6.3 Quadrilateral Elements, \(\underline{Q}_{k} - P_{k-1}\) Elements
We now discuss the stability and convergence of a family of quadrilateral elements. The lowest order of this family, the \(\underline{Q}_{2} - P_{1}\) element, is one of the most popular Stokes elements. These elements are discontinuous pressure elements and they originate from attempts to use the reduced integration technique which will be analysed in Sect. 8.12. Let us first consider what would appear to be a natural construction. Given k ≥ 1, the discrete spaces are defined as follows:
For k = 1, this yields the unstable \(\underline{Q}_{1} - P_{0}\) which will be considered in detail in Sect. 8.10. For \(k = 2\), we have the \(\underline{Q}_{2} - Q_{1}\) element which appears quite naturally in the use of reduced integration penalty methods (see [60]). This element is not stable and suffers from the same problems as the \(\underline{Q}_{1} - P_{0}\) element.
Let us now consider, instead of (8.6.6),
Using P k pressure instead of Q k is not a natural choice, although it is the good one. If the mesh is built of rectangles, the stability proof is an immediate consequence of Proposition 8.5.9, since (8.5.28) is satisfied for V h (indeed, the \(\underline{Q}_{2} - P_{0}\) is a stable Stokes element, see Remark 8.4.3). In the case of a general quadrilateral mesh, things are not so easy; even the definition of the space Q h is not so obvious and there have been different opinions, during the years, about two possible natural definitions. Following [90], we discuss in detail the case k = 2. We shall see that in this case there are important issues related to the approximation properties of finite elements on non-affine meshes.
8.6.3.1 The \(\underline{Q}_{2} - P_{1}\) Element
This element was apparently discovered around a blackboard at the Banff Conference on Finite Elements in Flow Problems (1979). Two different proofs of stability can be found in [223] and [351] for the rectangular case. This element is a relatively late comer in the field; the reason for this is that, as we stated earlier, using a P 1 pressure on a quadrilateral is not a standard procedure. It appeared as a cure for the instability of the \(\underline{Q}_{2} - Q_{1}\) element which appears quite naturally in the use of reduced integration penalty methods (see [60]). Another cure can be obtained by adding internal nodes (see [199]).
On a general quadrilateral mesh, the space Q h can be defined in two different ways: either Q h consists of (discontinuous) piecewise linear functions, or it is built by considering three linear shape functions on the reference unit square and mapping them to the general elements like it is usually done for continuous finite elements (see (2.1.59)). We point out that since the mapping F K from the reference element \(\hat{K}\) to the general element K in this case is bilinear but not affine, the two constructions are not equivalent. We shall refer to the first possibility as the unmapped pressure approach and to the second one as the mapped pressure approach.
In order to analyse the stability of either scheme, we use the macro-element technique presented in Sect. 8.5.4 with macro-elements consisting of one single element.
The unmapped pressure approach yields the original proof presented in [351]. Let M be a macro-element and \(q_{h} = a_{0} + a_{x}x + a_{y}y \in Q_{0,M}\) an arbitrary function in K M . If b(x, y) denotes the bi-quadratic bubble function on K, then \(\underline{v}_{h} = (a_{x}b(x,y),0)\) is an element of V 0, M and
implies a x = 0. In a similar way, we get a y = 0 and, since the average of q h on M vanishes, we have the macro-element condition q h = 0.
We now consider the mapped pressure approach, following the proof presented in [90]. There, it is recalled that the macro-element condition (8.5.24) can be related to an algebraic problem in which we are led to prove that a two-by-two matrix is non-singular. Actually, it turns out that the determinant of such a matrix is a multiple of the Jacobian determinant of the function mapping the reference square \(\hat{K}\) onto M, evaluated at the barycentre of \(\hat{K}\). Since this number must be non-zero for any element of a well-defined mesh, we can deduce that the macro-element condition is also satisfied in this case, and we can then conclude that the stability holds thanks to Proposition 8.5.6.
So far, we have shown that both the unmapped and the mapped pressure approach give rise to a stable Q 2 − P 1 scheme. However, as a consequence of the results proved in [20], we have that the mapped pressure approach cannot achieve optimal approximation order. Namely, the unmapped pressure space provides a second order convergence in L 2, while the mapped one achieves only O(h) in the same norm. In [90], several numerical experiments have been reported, showing that on general quadrilateral meshes (with constant distortion), the unmapped pressure approach provides a second order convergence (for both velocity in H 1 and pressure in L 2), while the mapped approach is only sub-optimally first order convergent. It is interesting to remark that, in this case also, the convergence of the velocities is suboptimal, according to the error estimate (8.2.17).
8.7 Three-Dimensional Stable Elements
Many elements presented in Sect. 8.6 have a three-dimensional extension. Some of them are schematically plotted in Fig. 8.7. However, there are important differences between the two and three-dimensional cases. One is that bubbles are at least of fourth degree in the three-dimensional case. Another difference is that the \(\underline{P}_{2} - P_{0}\) element is not stable: to control piecewise constant pressure, we need some degrees of freedom on the faces. It would indeed be possible to prove that the \(\underline{P}_{3} - P_{0}\) is stable but this yields a highly unbalanced approximation.
Some stable three-dimensional Stokes elements: (a) the MINI element, (b) the Crouzeix–Raviart element, (c) the \(\underline{Q}_{2} - P_{1}\) element
8.7.1 Continuous Pressure 3-D Elements
The most important continuous pressure element is the Hood-Taylor element, and its generalisations, which will be presented in the next section.
The families associated with the MINI element introduced in (8.6.1) and (8.6.2) can be generalised to the three-dimensional with an appropriate choice of bubbles. Consider a regular sequence of decompositions of Ω into tetrahedra.
Proposition 8.7.1.
The following pair of spaces are stable for any k ≥ 1
The proof follows easily, like in the 2D case, from Corollary 8.5.1 . □
The first member of the first family is the MINI element and the first member of the second family is a version of the Hood-Taylor element where the velocities are enriched by quartic bubbles. Paradoxically, this may increase the precision on pressure through a better inf-sup constant.
8.7.2 Discontinuous Pressure 3-D Elements
As we stated earlier, the situation for discontinuous pressure elements is less favourable than in the two-dimensional case. We first consider an example.
Example 8.7.1.
The SMALL element is the smallest three-dimensional one if one wants to use piecewise constant pressure. It is the analogue of the two-dimensional construction of Remark 8.4.2, but we now have to work on the faces and not on the edges. Let us thus consider, on a tetrahedral element, the cubic bubble \(\underline{b}_{3,F}\) associated to the face F = (i, j, k) and defined, using barycentric coordinates, by
We define our new space by adding on each face of a standard piecewise linear element such a cubic bubble. We shall denote this extra space on an element K by
The final space is thus
This provides the control of the flux on the face and one can easily check that we have stability for piecewise constant pressure.
Remark 8.7.1.
In fact, the bubble on the face is needed only for the normal component of the velocity. This implies some complexity for the implementation but greatly reduces the global number of degrees of freedom. We could thus use, instead of (8.7.4), denoting \(\underline{n}_{F}\) the normal to the face,
and \(V _{h} := ({(\mathcal{L}_{1}^{1})}^{3} + BF_{3}^{n}) \cap V.\) □
We must retain that discontinuous pressure elements in 3D require third degree polynomials. □
Example 8.7.2 (3D analogues of the Crouzeix–Raviart element).
In order to generalise the Crouzeix-Raviart element, we must first get a stable element for constant pressure. The previously defined SMALL element provides only first order accuracy. We therefore start from a quadratic approximation and enrich it by face bubbles to control the fluxes on the faces and by internal bubbles to control linear pressure. This yields
The stability is an easy consequence of Proposition 8.5.9.
The face bubbles do not increase the order of accuracy and we could employ the normal bubbles of (8.7.5) However, if one wants to stick to more standard elements, the natural thing would be to start from the stable \(\underline{P}_{3} - P_{0}\) element. To get a balanced precision, we define
We would then have third order accuracy but to the price of a quite large number of degrees of freedom. This can obviously be extended to higher degrees. □
Example 8.7.3 (Nonconforming elements).
A possibility to reduce the order of polynomials needed to obtain stable elements is to use nonconforming elements. The triangular \(P_{1}^{NC} - P_{0}\) easily generalises to tetrahedra in 3D. Also in this case, since \(\text{div}V _{h} \subset Q_{h}\), the restriction of the discrete solution to every element is truly divergence-free. The problems of coercivity are still there.
However, it is also possible to obtain second order without this problem. We have already seen in Remark 8.6.1 that one could get a variant of the Crouzeix-Raviart element using nonconforming bubbles. The construction can be extended to the three-dimensional case. Indeed, one can replace the face-bubble of (8.7.3) by its nonconforming version of (2.2.39), that is,
The internal bubble is also replaced by \((\lambda _{i}^{2} +\lambda _{ j}^{2} +\lambda _{ k}^{2} +\lambda _{ l}^{2}) - 1\) and we now manipulate only second degree polynomials, which is definitely an advantage. However, things are a little more complicated: we have too many degrees of freedom and one must remove the vertices. We refer to [203] for details. □
Example 8.7.4 (Quadrilateral \(\underline{Q}_{k} - P_{k-1}\) elements).
Given a mesh of hexahedra, we define
for k ≥ 2. We refer to the two-dimensional case (see Sect. 8.6.3) for the definition of the pressure space. In particular, we recall that Q h on each element consists of true polynomials and is not defined via the reference element. With the correct definition of the pressure space, the proof of stability for this element is a simple generalisation of the corresponding two-dimensional version. □
8.8 \(\underline{P}_{k} - P_{k-1}\) Schemes and Generalised Hood–Taylor Elements
The main result of this section (see Theorems 8.8.1 and 8.8.2) consists in showing that a family of popular Stokes elements satisfies the inf-sup condition (8.2.16). The first element of this family has been introduced in [249] and for this reason, the members of the whole family are usually referred to as generalised Hood–Taylor elements.
This section is organised in two subsections. In the first one, we discuss discontinuous pressure approximations for the \(\underline{P}_{k} - P_{k-1}\) element in the two-dimensional triangular case; it turns out that this choice is not stable in the lower order cases and requires suitable conditions on the mesh sequences for the stability of the higher order elements.
The last subsection deals with the generalised Hood–Taylor elements, which provide a continuous pressure approximation in the plane (triangles and quadrilaterals) and in the three-dimensional space (tetrahedra and hexahedra).
8.8.1 Discontinuous Pressure \(\underline{P}_{k} - P_{k-1}\) Elements
In this subsection, we shall recall the statement of a basic result by Scott and Vogelius (see [347]) which, roughly speaking, says: under suitable assumptions on the decomposition \(\mathcal{T}_{h}\) (in triangles), the pair \(V _{h} := {(\mathcal{L}_{k}^{1})}^{2} \cap V\), \(Q_{h} := \mathcal{L}_{k-1}^{0} \cap Q\) satisfies the inf-sup condition for k ≥ 4.
On the other hand, the problem of finding stable lower order approximations has been studied by Q in [328], where interesting remarks are made on this scheme and where the possibility of filtering out the spurious pressure modes is considered.
In order to state in a precise way the restrictions that have to be made on the triangulation for higher order approximations, we assume that Ω is a polygon, and that its boundary ∂ Ω has no double points. In other words, there exist two continuous piecewise linear maps x(t), y(t) from [0, 1[ into \(\mathbb{R}\) such that
Clearly, we will have \(\lim \limits _{t\rightarrow 1}x(t) = x(0)\) and \(\lim \limits _{t\rightarrow 1}y(t) = y(0)\). We remark that we are considering a less general case than the one treated by Scott and Vogelius [347]. We shall make further restrictions in what follows, so that we are actually going to present a particular case of their results.
Now let V be a vertex of a triangulation \(\mathcal{T}_{h}\) of Ω and let \(\theta _{1},\ldots ,\theta _{n}\), be the angles, at V , of all the triangles meeting at V , ordered, for instance, in the anticlockwise sense. If V is an internal vertex, we also set \(\theta _{n+1} :=\theta _{1}\). Now we define S(V ) according to the following rules.
It is easy to check that S(V ) = 0 if and only if all the edges of \(\mathcal{T}_{h}\) meeting at V fall on two straight lines. In this case, V is said to be singular [347]. If S(V ) is positive but very small, then V will be “almost singular”. Thus, S(V ) measures how close V is to be singular.
We are now able to state the following result.
Proposition 8.8.1 ([347]).
Assume that there exist two positive constants c and δ such that
and
Then, the choice \(V _{h} = {(\mathcal{L}_{k}^{1})}^{2} \cap V,\;Q_{h} = \mathcal{L}_{k-1}^{0} \cap Q,\;k \geq 4\) , satisfies the inf-sup condition with a constant depending on c and δ but not on h. □
Remark 8.8.1.
Condition (8.8.6) is worth a few comments. The trouble is that S(V ) = 0 makes the linear constraints on \(\underline{u}_{h}\), arising from the divergence-free condition, linearly dependent (see, also, Examples 8.10.2 and 8.10.3). When this linear dependence appears, some part of the pressure becomes unstable. However, we have met this situation in Example 8.10.3 and this was in fact the key to convergence, provided a condition on data was fulfilled. The same analysis would hold here and the unstable part of pressure could be filtered out. □
Remark 8.8.2.
The \(\underline{P}_{k} - P_{k-1}\) element can obviously be stabilised by adding bubbles to the velocity space in the spirit of Sect. 8.5.5 (see Proposition 8.5.9). For a less expensive stabilisation, consisting in adding bubbles only in few elements, see [72]. □
8.8.2 Generalised Hood–Taylor Elements
In this subsection, we recall the results proved in [71, 73] concerning the stability of the generalised Hood–Taylor schemes. On triangles or tetrahedra, velocities are approximated by a standard \(\underline{P}_{k}\) element and pressures by a standard continuous P k − 1, that is, \(\underline{v}_{h} \in {(\mathcal{L}_{k}^{1})}^{n} \cap V\) (n = 2, 3), \(p \in \mathcal{L}_{k-1}^{1} \cap Q\). This choice has an analogue on rectangles or cubes using a \(\underline{Q}_{k}\) element for velocities and a Q k − 1 element for pressures. The lowest order triangular element (i.e., k = 2) has been introduced by Hood and Taylor in [249]. Several papers are devoted to the analysis of this popular element.
The degrees of freedom of some elements belonging to this family are reported in Fig. 8.8.
Some stable elements belonging to the Hood–Taylor family
Remark 8.8.3.
Another element that has been used because of the simplicity of its shape functions is the so-called \((\underline{P}_{1} -\mathit{iso} -\underline{ P}_{2}) - P_{1}\) element, It is sketched in Fig. 8.9. It is a composite element assembled from four piece-wise linear elements for velocity while pressure remains linear on the macro-element. The same technique of proof that yields stabiliy of the classical Hood-Taylor element could be used to show the inf-sup condition for this composite element. □
The \((\underline{P}_{1}\mathit{iso}\underline{P}_{2}) - P_{1}\) element
The first proof of convergence was given for the two-dimensional case in [61] where a weaker form of the inf-sup condition was used. The analysis was subsequently improved in [375], who showed that the classical inf-sup condition is indeed satisfied (see Verfürth’s trick in Sect. 8.5.2). The macro-element technique can easily be used for the stability proof of the rectangular and cubic element (of any order) as well as for the tetrahedral case when k = 2 (see [351]). In [122], an alternative technique of proof has been presented for the triangular and tetrahedral cases when k = 2. This proof generalises to the triangular case when k = 3 (see [121]). Finally, a general proof of convergence can be found in [71] and [73] for the triangular and tetrahedral case, respectively.
We now state and prove the theorem concerning the two-dimensional triangular case (see [71]).
Theorem 8.8.1.
Let Ω be a polygonal domain and \(\mathcal{T}_{h}\) a regular sequence of triangular decompositions of it. Then, the choice \(V _{h} := {(\mathcal{L}_{k}^{1} \cap H_{0}^{1}(\Omega ))}^{2}\) and \(Q_{h} := \mathcal{L}_{k-1}^{1} \cap L_{0}^{2}(\Omega )\) satisfies the inf-sup condition (8.2.16) for any k ≥ 2 if and only if each triangulation contains at least three triangles.
Proof (Step 1: necessary part).
Let us show first that the hypothesis on the mesh is necessary. If \(\mathcal{T}_{h}\) only contains one element, then it is easy to see that the inf-sup constant is zero (otherwise, it should be \(\text{div}V _{h} \subset Q_{h}\), which is not the case since the functions in Q h are not zero at the vertices). We shall show that if \(\mathcal{T}_{h}\) contains only two triangles T 1 and T 2, then there exists one spurious pressure mode. This implies that, also in this case, the inf-sup constant vanishes. We choose the coordinate system (x, y) in such a way that the common edge of T 1 and T 2 lies on the y-axis. Moreover, we suppose that T 2 is the reference triangle and T 1 the symmetric one with respect to the x-axis (see Fig. 8.10). The general case can then be handled by means of suitable affine mappings. We denote by λ i, a and λ i, b the barycentric coordinates relative to the vertices a and b, respectively, belonging to the element T i , i = 1, 2. It is easy to check that it holds: \(\lambda _{1,a} = 1 + x - y\), λ 1, b = y, \(\lambda _{2,a} = 1 - x - y\), and λ 2, b = y. We shall also make use of the function λ 2, c = x. Let L(x) be the Legendre polynomial of degree k − 2 on the unit interval with respect to the weight \(w(x) = x{(1 - x)}^{3}\) and consider the function p(x) ∈ Q h defined as follows:
We shall show that \(\mathop{\mathrm{g\underline{ra}d}}\nolimits p\) is orthogonal to any velocity \(\underline{v} \in V _{h}\). Since p does not depend on y, we can consider the first component v 1 of \(\underline{v}\) only, which, by virtue of the continuity at x = 0 and of the boundary conditions, has the following general form:
where the subscripts denote the degrees of the polynomials A, B and C. We then have
where q(x, y) is a polynomial of degree k − 3 and where the term involving C disappears by virtue of the symmetries. The last integral reads
and an explicit calculation shows that Q(x) is of the form
where p k − 3 is a polynomial of degree k − 3. We can now conclude with the final computation
The reference triangle and its symmetric
Step 2: sufficient part. The idea of the proof consists in considering, for each h, a partition of the domain Ω in sub-domains containing exactly three adjacent triangles. By making use of Proposition 8.5.4 and the technique presented in Sect. 8.5.1, it will be enough to prove the inf-sup condition for a single macro-element, provided we are able to bound the number of intersections between different sub-domains (basically, every time two sub-domains intersect each other, a factor \(1/\sqrt{2}\) shows up in front of the final inf-sup constant). Indeed, it is possible to prove that, given a generic triangulation of a polygon, it can be presented as the disjoint union of triplets of triangles and of polygons that can be obtained as unions of triplets with at most three intersections.
Given a generic macro-element a ′ ∪b ′ ∪c ′, consider the (x, y) coordinate system shown in Fig. 8.11, so that the vertices are B ′ = (0, 0), D ′ = (1, 0), E ′ = (α, β). By means of the affine mapping \(x\prime = x +\alpha y,\) y ′ = β y, the Jacobian of which is β, we can consider the macro-element a ∪b ∪c shown in Fig. 8.12, so that b is the unit triangle. Since β≠0, the considered affine mapping is invertible. With an abuse of notation, we shall now denote by Ω the triplet a ∪b ∪c and by V h and Q h the finite element spaces built on it.
A generic triplet of triangles
A macro-element where b is the reference triangle
We denote by \(\lambda _{\mathit{AB}}^{a}\) the barycentric coordinate of the triangle a vanishing on the edge A B (analogous notation holds for the other cases). Moreover, we denote by \(L_{i,x}^{a}(x)\) the i-th Legendre polynomial in [x A , 0], with respect to the measure μ a, x defined by
where x A is the x-coordinate of the vertex A. We shall make use of the following Legendre polynomials, which are defined in a similar way: \(L_{i,x}^{b}\) (its definition involves \(\lambda _{\mathit{ED}}^{b}\) and \(\lambda _{\mathit{BD}}^{b}\)), \(L_{i,y}^{b}\) (using \(\lambda _{\mathit{BE}}^{b}\) and \(\lambda _{\mathit{BD}}^{b}\)), and \(L_{i,y}^{c}\) (using \(\lambda _{\mathit{BC}}^{c}\) and \(\lambda _{\mathit{CD}}^{c}\)).
Standard properties of the Legendre polynomials ensure that we can normalise them, for instance, by requiring that they assume the same value (say 1) at the origin. We now prove by induction with respect to the degree k that a modified inf-sup condition holds true (see Verfürth’s trick in Sect. 8.5.2). Namely, for any q h ∈ Q h , we shall construct \(\underline{v}_{h} \in V _{h}\) such that
The case k=2. This is the original Hood–Taylor method. Given p ∈ Q h , we define \(\underline{v}_{h} = (v_{1}(x,y),v_{2}(x,y))\), triangle by triangle, as follows:
where the quantities σ and τ are equal to ± 1 so that the expressions
are non-negative. First of all, we observe that \(\underline{v}_{h}\) is an element of V h : its degree is at most two in each triangle, it vanishes on the boundary and it is continuous across the internal edges because so is the tangential derivative of p.
It is easy to check that \(\vert \vert \underline{v}_{h}\vert \vert _{0} \leq c_{1}\vert \vert \mathop{\mathrm{g\underline{ra}d}}\nolimits p\vert \vert _{0}\). In order to prove the first equation in (8.8.14), we shall show that the quantity\(\vert \vert \vert \mathop{\mathrm{g\underline{ra}d}}\nolimits p\vert \vert \vert = -\int _{\Omega }\underline{v}_{h} \cdot \mathop{\mathrm{g\underline{ra}d}}\nolimits p\) vanishes only when \(\mathop{\mathrm{g\underline{ra}d}}\nolimits p\) is zero. From the equality
it follows that
These last equations, together with the fact that each component of \(\mathop{\mathrm{g\underline{ra}d}}\nolimits p\) is constant if p ∈ Q h , easily imply that
The case k > 2. Given p in Q h , if p is locally of degree k − 2, then the result follows from the induction hypothesis. Otherwise, there exists at least one triangle of Ω in which p is exactly of degree k − 1. Like in the previous case, we define \(\underline{v}_{h} = (v_{1}(x,y),v_{2}(x,y))\) as follows:
with the same assumption on σ and τ, so that the terms
are non-negative. The same arguments as for k = 2, together with the described normalisation of the Legendre polynomials, show that \(\underline{v}_{h}\) belongs to V h .
In order to conclude the proof, we need to show that if\(\vert \vert \vert \mathop{\mathrm{g\underline{ra}d}}\nolimits p\vert \vert \vert = -\int _{\Omega }\underline{v}_{h} \cdot \mathop{\mathrm{g\underline{ra}d}}\nolimits p = 0\), then the degree of \(\mathop{\mathrm{g\underline{ra}d}}\nolimits p\) is strictly less than k − 2. As before, \(\vert \vert \vert \mathop{\mathrm{g\underline{ra}d}}\nolimits p\vert \vert \vert = 0\) implies
The last equalities imply
and
It follows that the degree of \(\mathop{\mathrm{g\underline{ra}d}}\nolimits p\) is strictly less than k − 2 in contrast to our assumption. □
Remark 8.8.4.
The proof of the theorem shows that the continuity hypothesis on the pressure space Q h can be weakened up to require that q h is only continuous on triplets of elements. □
We conclude this subsection by stating the three-dimensional analogue to the previous theorem and by recalling the main argument of the proof presented in [73].
Theorem 8.8.2.
Let Ω be a polyhedral domain and \(\mathcal{T}_{h}\) a regular sequence of decompositions of it into tetrahedra. Assume that every tetrahedron has at least one internal vertex. Then, the choice \(V _{h} := {(\mathcal{L}_{k}^{1} \cap H_{0}^{1}(\Omega ))}^{3}\) and \(Q_{h} := \mathcal{L}_{k-1}^{1} \cap Q\) satisfies the inf-sup condition (8.2.16) for any k ≥ 2.
Proof.
We shall make use of the macro-element technique presented in Sect. 8.5.4. In particular, we shall use Proposition 8.5.7 and the comments included in Remark 8.5.6.
We consider an overlapping macro-element partition of \(\mathcal{T}_{h}\) as follows: for each internal vertex x 0, we define a corresponding macro-element \(M_{x_{0}}\) by collecting all elements which touch x 0. Thanks to the regularity assumptions on the mesh, we only have to show that the macro-element condition (8.5.24) holds true (see, in particular, Remark 8.5.6).
Let us consider an element \(K \in M = M_{x_{0}}\) and an edge e of K which touches x 0. With a suitable choice of the coordinate system, we can suppose that the direction of e coincides with that of the x axis. With the notation of Sect. 8.5.4, we shall show that a function in K M cannot contain functions which depend on x in K. Namely, given a function p ∈ Q 0, M , we can define a function \(\underline{v} \in V _{0,M}\) as follows:
where K i is a generic element of M sharing the edge e with K and λ j, i , j = 1, 2, are the barycentric coordinates of K i associated with the two faces of K i which do not touch e. On the remaining elements, each component of \(\underline{v}\) is set equal to zero. It is clear that \(\underline{v}\) is a k-th order polynomial in K i and, since p is continuous in M, ∂ p ∕ ∂ x is continuous across the faces which meet at e and the function \(\underline{v}\) is continuous as well. Hence, \(\underline{v}\) belongs to V 0, M .
From the definition of Q 0, M , it turns out that
The last relation implies that p does not depend on x in K i for any i and, in particular, in K. On the other hand, we can repeat the same argument using as e the other two edges of K meeting at x 0 and, since the directions of the three used edges are independent, we obtain that p is constant in K. □
Remark 8.8.5.
From the previous proof, we can deduce that the hypotheses on the triangulation can be weakened, by assuming that each tetrahedron has at least three edges which do not lie on the boundary of Ω and which are not in the same plane. On the other hand, given a generic mesh of tetrahedra, it is not difficult to add suitable elements in order to meet the requirements of the previous theorem. □
Remark 8.8.6.
The main argument in the proof of the previous theorem is the straightforward generalisation of the two-dimensional case. Indeed, the proof of Theorem 8.8.1 could be carried out using the macro-element technique as well. □
8.9 Other Developments for Divergence-Free Stokes Approximation and Mass Conservation
From the discussion presented so far, it is clear that, in general, the incompressibility constraint \(\mathrm{div}\underline{u} = 0\) is not satisfied exactly at the discrete level. More precisely, the discrete velocity field \(\underline{u}_{h}\) fulfills the following equation
so that the equality \(\mathrm{div}\underline{u}_{h} = 0\) holds in general only if
Almost all stable elements that we have presented up to now do not satisfy (8.9.1), the only exception being the two dimensional Scott–Vogelius scheme \(\underline{P}_{k} - P_{k-1}\), which however requires severe mesh restrictions (see Sect. 8.8.1). Another example is presented in Example 8.10.3, On the other hand, discrete schemes that fail to satisfy the divergence-free condition (at least locally) can lead to undesired instabilities when used for the resolution of more complex problems. This is the case, for instance, when a Stokes solver is used for the approximation of non linear problems (see [39]), or when the incompressibility condition is related to a physical mass conservation property, like in fluid-structure interaction problems (see [78, 280]).
For this reason, a very active research area concerns investigations trying to develop divergence-free Stokes elements, at least in a local sense.
8.9.1 Exactly Divergence-Free Stokes Elements, Discontinuous Galerkin Methods
The simplest idea in order to satisfy (8.9.1) is to use a C 1 approximation of the velocity field and to take as space of pressures exactly Q h = div(V h ). Here we do not follow this approach, but we focus on suitably chosen mixed approximations of the Stokes problem.
Early attempts to develop divergence-free finite elements for the approximation of the Stokes problem made use of particular mesh sequences. Besides the already mentioned Scott–Vogelius family (see Sect. 8.8.1), a two dimensional approximation involving a mesh of rectangles has been introduced in [253, 385]. The lowest element of the family is constructed as follows: \(V _{h} = {\mathcal{L}}^{1}(P_{2,1} \times P_{1,2},\mathcal{T}_{h})\), \(Q_{h} = \mathrm{div}(V _{h}) \subset \mathcal{L}_{[1]}^{1}\). It is clear that the use of rectangular elements imposes limitations to the geometry of the domain Ω, which make the scheme unappealing for practical applications.
8.9.1.1 Discontinuous Galerkin Approximations
A more interesting approach arises from the use of discontinuous Galerkin approximations. A first possibility is to use a completely discontinuous finite element space for the approximation of the velocity together with a postprocessing procedure (see [156]), or H(div) conforming elements in order to avoid the postprocessing (see [157]).
A more recent approach is based on the idea of using H(div) conforming elements for the approximation of the velocities and to enrich them in order to obtain the stability (see [238]); the enrichment is performed locally by means of divergence-free polynomials (defined as the curl of suitably chosen bubble functions), so that the scheme remains conservative. The construction holds on simplicial meshes in two and three dimensions and is based on \(\mathcal{B}\mathcal{D}\mathcal{M}\) and \(\mathcal{R}\mathcal{T}\) spaces. This research is further improved in [237], where a conforming divergence-free element, which can be implemented on two dimensional triangular meshes, is presented. In this case the enrichment is based on rational functions which ensure stability and modify the tangential components of the basis functions across the interelements in order to guarantee their continuity. Some a posteriori error estimators (Sect. 7.11) for these methods have been considered in [251].
8.9.2 Stokes Elements Allowing for Element-Wise Mass Conservation
From what we have seen, it is not so easy to obtain a discrete velocity with vanishing divergence pointwise. On the other hand, in several applications it might be desirable to have a local (element-wise) conservation of mass. From (8.9.1) it is clear that discontinuous pressure schemes enjoy automatically a local conservation property. In particular, if Q h contains piecewise constants, then \(\mathrm{div}\underline{u}_{h}\) has zero mean value on each element. In this respect, we believe that the \(\underline{Q}_{2} - P_{1}\) scheme (see Sect. 8.6.3.1) is one of the best performing method for quadrilateral meshes. For simplicial meshes, the SMALL element of Example 8.7.1 provides what seems to be the simplest element ensuring local mass conservation.
Non conforming schemes can also achieve this goal. In particular the use of non conforming piecewise linear element for the velocity and piecewise constants for the pressures yields a simple locally divergence-free scheme. (see [165] and Sect. 8.4.4 for a discussion about this method). The extension to quadrilateral meshes requires a careful choice of the non conforming space (see [330]).
For continuous pressure schemes, however, the situation is more complicate. Relation (8.9.1), in particular, shows that the discrete divergence-free condition has to be considered in a non local sense. For this reason, there have been studies trying to modify standard spaces in order to achieve a more local conservation of mass. The main idea behind this technique consists in adding piecewise constants to the pressure space. It is clear that this modification allows for a local mass conservation (actually, the method is transformed into a scheme with discontinuous pressures), but can work only if it does not affect the validity of the inf-sup condition: a larger pressure space is indeed a potential source of trouble for the stability.
Indeed, it can be shown that generalised Hood–Taylor (see Sect. 8.8) can be modified by adding piecewise constants to the pressure space and preserving the inf-sup condition (k ≥ 2 in two dimensions and k ≥ 3 in three dimensions). The same procedure can be applied to the \(\underline{P}_{1}\)iso\(\underline{P}_{2} - P_{1}\) element. For the Hood–Taylor scheme, the idea was suggested in [231, 232, 369], where numerical evidence of the improvement was given (see also [144]). The proof of the stability of the enhanced lowest order Hood–Taylor scheme for triangular and rectangular meshes, can be found in [322, 329, 364]. A more comprehensive discussion, including a general proof of stability, can be found in [83].
8.10 Spurious Pressure Modes
As we stated in the introduction to this chapter, the approximation of the Stokes problem has been developed mostly independently of the theory of mixed methods. This led to the use of some approximations which did not satisfy the inf-sup conditions and which generated strange results, specially for the pressure components. This generated the concept of spurious pressure modes.
For the Stokes problem with Dirichlet boundary conditions, pressure is defined up to a constant which is the kernel of the gradient operator. This is a natural pressure mode. However, this mode may not be the only one in discrete problems. For a given choice of V h and Q h , we define the space S h of spurious pressure modes as follows:
It is clear that a necessary condition for the validity of the inf-sup condition (8.2.16) is the absence of spurious modes, that is,
In particular, if S h is non-trivial, then the solution p h to the discrete Stokes problem (8.2.11) can be changed to p h + s h , which is still a solution when s h ∈ S h . Spurious modes correspond to null singular values as discussed in Sect. 5.6.2. The existence of spurious modes is in many cases strongly mesh dependent. They will appear on special regular meshes but will remain if such meshes are slightly distorted.
We shall illustrate how this situation may occur with the following example.
Example 8.10.1 (The Q 1 − P 0 element).
Among quadrilateral elements, the \(\underline{Q}_{1} - P_{0}\) element is the first that comes to mind. It is defined as (see Fig. 8.13):
This element is strongly related, for rectangular meshes, to some finite difference methods [206]. Its first appearance in a finite element context seems to be in [255].
However simple it may look, the \(\underline{Q}_{1} - P_{0}\) element is one of the hardest elements to analyse and many questions are still open about its properties. This element does not satisfy the inf-sup condition: it strongly depends on the mesh. For a regular mesh, the space of spurious modes is one-dimensional. More precisely, \(\mathop{\mathrm{g\underline{ra}d}}\nolimits _{h}q_{h} = 0\) implies that q h is constant on the red and black cells if the mesh is viewed as a chequerboard (Fig. 8.14).
The chequerboard mode
The \(\underline{Q}_{1} - P_{0}\) element
This means that one singular value (cf. Chap. 3.4.3) of the operator B h = div h is zero. Moreover, it has been checked by computation [286] that a large number of positive singular values converge to zero when h becomes small. In [263], it has indeed been proved that the second singular value is O(h) and is not bounded below (see also [314]). The \(\underline{Q}_{1} - P_{0}\) element has been the subject of a vast literature. We shall come back to it in Sect. 8.10.2. □
We shall now present a few more examples and distinguish between local and global spurious pressure modes.
Example 8.10.2 (The criss-cross \(\underline{P}_{1} - P_{0}\) element).
Let us consider a mesh of quadrilaterals divided into four triangles by their diagonals (Fig. 8.2). We observed, in Example 8.3.2, that the \(\underline{P}_{1} - P_{0}\) element, on general meshes, is affected by locking, that is, the computed velocity vanishes. On the mesh introduced above, however, it is easy to see that non-zero divergence-free functions can be obtained. The divergence is constant on each triangle. This means that there are four linear relations between the values of the partial derivatives. It is easily seen that one of them can be expressed as a combination of the others, this fact being caused by equality of tangential derivatives along the diagonals. To make things simpler, we consider the case where the diagonals are orthogonal (Fig. 8.15) and we label by A, B, C, D the four triangles. We then have, by taking locally the coordinate axes along the diagonals and by denoting by \(\underline{{u}}^{K}\) the restriction of a function of V h to the element K,
On the other hand, one has at the point M
It is easy to check that this makes one of the four conditions (8.10.4) redundant. The reader may check the general case by writing the divergence operator in a non orthogonal coordinate system.
The reference criss-cross
The consequence of the above discussion is that on each composite quadrilateral, one of the four constant pressure values will be undetermined. The dimension of \(\mathrm{Ker}B_{h}^{t}\) will be at least as large as the number of quadrilaterals minus one. This is what we shall call local modes.
Thus, three constraints remain on each composite quadrilateral element. If we admit that two of them can be controlled, using the methods of Sect. 8.5.5, by the “internal” node M, we obtain an element that is very similar to the \(\underline{Q}_{1} - P_{0}\) element with respect to the degrees of freedom. Indeed, it can be checked that on a regular mesh, an additional, global chequerboard mode occurs and that the behaviour of this approximation is essentially the same as that of the \(\underline{Q}_{1} - P_{0}\) element that will be discussed in details in Sect. 8.10.2. These analogies have been pointed out, for instance, in [82]. □
The above example has shown the existence of two kinds of spurious pressure modes. In the case of the criss-cross \(\underline{P}_{1} - P_{0}\) element presented in the previous example, dimS h grows as h goes to 0 and there exists a basis of S h with local support (that is, the support of each basis function can be restricted to one macro-element). We shall refer to these modes as local spurious modes. Such pressure modes can be eliminated by considering a composite mesh (in the previous example a mesh of quadrilaterals instead of triangles) and using a smaller space for the pressures by deleting some degrees of freedom from the composite elements. If the original space is to be employed, one must check the extra compatibility conditions. This can often be done by a small change in the data. This will be the case in Example 8.10.3.
If we now consider the \(\underline{Q}_{1} - P_{0}\) example (see Example 8.10.1), the dimension of S h does not grow when h goes to 0 and no basis can be found with a local support. We then have a global spurious mode which cannot be eliminated as easily as the local ones. Global modes usually appear on special (regular) meshes and are symptoms that the behaviour of the element at hand is strongly mesh dependent and requires a special care. Some elements may generate both local and global modes as we have seen in the criss-cross \(\underline{P}_{1} - P_{0}\) method (see Example 8.10.2).
Example 8.10.3 (The criss-cross \(\underline{P}_{2} - P_{1}\) element).
Another simple example where a local mode occurs is the straightforward extension of the previous example to the case of a \(\underline{P}_{2} - P_{1}\) approximation. This element has an interest because it is the simplest really divergence-free element, that is, KerB h ⊂ KerB. Unfortunately, its three-dimensional counterpart does not seem to exist. We consider, as in Fig. 8.16, a mesh of quadrilaterals divided into triangles by its diagonal. This means that on each quadrilateral, we have 12 discrete divergence-free constraints, and it is easily seen by the argument of Example 8.10.2, written at the point M, that one of them is redundant. Thus, one spurious mode will appear for each composite quadrilateral. However, in this case, no global mode will appear. The analysis of this element is also related to the work of [153] by considering the stream function associated with a divergence-free function. Considering the space of discrete pressures where the spurious modes are removed, a standard proof using internal degrees of freedom shows that one has a stable approximation. □
8.10.1 Living with Spurious Pressure Modes: Partial Convergence
The presence of spurious modes can be interpreted as a signal that the pressure space used is in some sense too rich. We therefore can hope to find a cure by using a strict subspace \(\hat{Q}_{h}\) of Q h as the space of the discrete pressures, in order to obtain a stable approximation. The question arises whether or not this stability can be used to prove at least a partial result on the original approximation. One can effectively get some results in this direction as discussed in Sect. 5.6.3. In general, we cannot make a direct use of the singular value decomposition but, in some cases, we can identify a guilty subspace.
The criss-cross \(\underline{P}_{2} - P_{1}\) element
We suppose, here, that Q and Q h can be identified to their dual, as it is indeed the case for the Stokes problem.
Following Sect. 5.3.3, we suppose that we know subspaces \(\hat{V }_{h}\) and \(\hat{Q}_{h}\) of V h and Q h such that the couple \(\hat{V }_{h} \times \hat{ Q}_{h}\) is stable. We denote \(\tilde{Q}_{h}\) the orthogonal complement of \(\hat{Q}_{h}\) in Q h . To apply the result of Sect. 5.3.3, we shall need to obtain the following:
We emphasise that this will be generally possible only on special meshes. We now make the hypothesis that in (8.1.1), g has no component in \(\tilde{Q}_{h}\), that is,
This condition is a restriction on admissible data. In practice, it will imply an extra regularity condition on g which will in turn enable us to obtain (8.10.7) through a small modification of g. We mean by small that this modification should not jeopardise the accuracy of the approximation. If we refer to Sect. 5.6.2, by supposing (8.10.7), we have killed the unstable part of \(\underline{u}_{h}\). On the other hand, p h will have components in \(\tilde{Q}_{h}\). However, the part of p h in \(\hat{Q}_{h}\) will be stable and will provide a reasonable approximation of the solution. More precisely, under these hypotheses, Proposition 5.3.1 yields the following results:
Example 8.10.4.
The simplest example is the case of Example 8.10.3. In this case, we have \(\hat{V }_{h} = V _{h}\). When the local modes are filtered, pressure will converge, provided g has no component in these modes. This implies a slight restriction of data. □
The most important case is, however, the \(\underline{Q}_{1} - P_{0}\) element which we discuss in the next section.
8.10.2 The Bilinear Velocity-Constant Pressure \(\underline{Q}_{1} - P_{0}\) Element
We now come back to a rapid analysis of what is probably (and unfortunately!) the most popular of all elements for incompressible materials. This is perhaps also the hardest to analyse and as we shall see, only partial results are known. Origins of this element can be traced back to finite difference methods [206] and its peculiar properties were soon recognised. In particular, the chequerboard pressure mode was already a familiar feature long before the scheme used were written in terms of finite elements.
Let us summarise the basic facts. On a regular mesh, for a problem with Dirichlet boundary conditions, two singular values of the matrix (cf. Sect. 5.6.2) vanish instead of one. We thus have apure spurious pressure mode in the terminology of [340, 341]. This spurious mode implies a compatibility condition on the data, which is, in most cases but not always, easily satisfied. When the mesh is slightly distorted, only one singular value is zero, corresponding to constants, but the second one is very small, as the zero has become some value depending on the mesh distortion, thus implying an ill-conditioning of the problem. In many computations, this ill-conditioning is fortunately almost restricted to pressure: we have what [340, 341] call an impure pressure mode which can be eventually filtered but often does not seem to affect (at least substantially) the computation of velocity. This is still not, however, the whole story. One could indeed hope from all this that an inf-sup stability condition could hold for the third singular value instead of the second and that we could have stability in a simple quotient space. Experimental evidence showed this hope to be false: on a regular mesh, a large number of eigenvalues converge to zero at order h [286]. Johnson and Pitkäranta [263] indeed proved the constant k h to be O(h) (see also [99, 100, 288, 314]). The standard estimates would then lead to the conclusion that no convergence will occur, in complete contradiction with experience. The paper of Johnson and Pitkäranta provided a first result by showing, on a regular mesh, that under stricter regularity assumptions than usual on the solution, convergence could take place.
Pitkäranta and Stenberg [324] proved a convergence result, without special regularity assumptions for a special type of mesh. We have already discussed, in Sect. 8.10.1, following Sect. 5.6.2, the underlying algebraical issues involved. If there is a “stable part”, the data corresponding to the unstable modes should be null or small. In this case, the velocity can indeed be expected to behave well but the pressure part is doomed. We shall now consider these results for our particular case. To make things simpler, we shall first consider the case of a regular rectangular mesh. On such a mesh, we consider a macro-element (Fig. 8.17) M formed of four quadrilaterals.
Macro-element and degrees of freedom
On this macro-element, a piecewise constant pressure has four degrees of freedom. We introduce a local basis on \(M,\;\phi _{1},\;\phi _{2},\;\phi _{3},\;\phi _{4}\) described symbolically on the Fig. 8.18.
Pressure basis functions on M
A chequerboard mode will obviously take its roots in ϕ 4. We therefore introduce quite naturally the space
which will be the stable part and
the unstable part. Stability of (V h , \(\hat{Q}_{h})\) is thus immediate from the standard techniques, building a B-compatible operator. In fact, we shall build it for a subspace \(\hat{V }_{h}\) of V h which will make it, a fortiori, valid for V h . The choice of \(\hat{V }_{h}\) can be inferred from other well-known elements. We now use, as degrees of freedom, the two values of velocity at the vertices of M and at its barycentre and the normal value (rather a correction to this value) at mid-side nodes. The tangential component is thus linear on each edge and determined by the values at the vertices (Fig. 8.17). To build a B-compatible operator, we set
To determine the normal node on the edges of M, we take
where \(\underline{\underline{n}}_{e}\) is the normal to e. This is enough to control the flux at interfaces and the part of pressure (ϕ 1, M ) which is constant on M is controlled. The ϕ 2. M and ϕ 3, M components are controlled by
It is not difficult to check that \(\Pi _{h}\underline{u} =\hat{\underline{ u}}_{h}\) is a B-compatible operator for \(\hat{V }_{h} \times \hat{ Q}_{h}\).
We would now want to apply Proposition 5.3.1. First, this implies a condition on data.
Remark 8.10.1.
Following Sect. 5.3.3, to get reasonable results, the data should satisfy
which means, here,
This in fact corresponds to a regularity condition. It is easy to check that on our macro-element, we have
This is enough to show that this integral can be made null through a small perturbation of data. □
In order to apply Proposition 5.3.1, we now need to check (5.3.19) that is now
It should be seen, in order to check this, that the shape function \(\underline{w}_{1}\) associated to vertex P 1, for instance, is a function of Q 1(M) having the whole of M as its support. A straightforward computation then shows that one has
In the same way, the shape function \(\underline{w}_{12}\) associated with node P 12 satisfies
and this is also true in the adjacent element because the mesh is aligned. The shape function associated with the barycentre trivially satisfies the condition. Condition (5.3.19) therefore holds and we have, by Proposition 5.3.1,
In the present case, it is clear that an error estimate in \(\hat{V }_{h}\) has the same order as an estimate in V h and the result is therefore almost optimal. We also have convergence of (filtered) pressure in \(\hat{Q}_{h}\) by estimate (5.3.25). Following [324], we can now extend this result to the case where the mesh is made from super macro-elements like in Fig. 8.19.
A super-macro S M and its sub-macros
A general quadrilateral is divided in a regular way into 16 quadrilaterals. It is well-known [202] that on a non rectangular mesh, at least a four by four patch of elements is needed to generate a non-trivial discrete divergence-free function. We thus have four “sub-macros” like in the previous case. The space of filtered pressures \(\hat{Q}_{h}\) is taken exactly as on the regular mesh and is still defined by (8.10.10). The space \(\tilde{V }_{h}\) is defined by the following degrees of freedom: the values of velocity at the vertices of the M i , the values at the barycentres of the M i and a correction of the component of velocity parallel to the mesh at the mid-side nodes of the M i internal to S M. One can directly build an interpolation operator enabling us to check the inf-sup condition. Mid-side nodes of SM control the part of pressure constant on the whole of S M. Internal mid-side nodes ensure mass-balance on each M i and the nodes at the barycentres of the M i end the job. It must be remarked that the alignment of mid-side velocities along the mesh is an essential feature of the construction.
In order to prove condition (5.3.18), the only hard point is to check that (8.10.18) still holds on every M i . We refer the reader to [324] for this proof. It is then possible to use Proposition 5.3.1 and to get optimal error estimates.
This is still not the whole story about this peculiar element. It is also possible to prove stability [276, 354] on meshes built from macro-elements like in Fig. 8.20 without filtering or using another subterfuge.
A special macro-element
This is coherent with the known experimental fact that on a general distorted mesh, pressure modes disappear and the inf-sup constant is independent of h. This last fact is still resisting analysis. It is our hope that the above technique could be generalised to yield the complete result.
The above discussion can be extended to the three-dimensional case. Things are made still more complicated by the fact that on a regular mesh (let say a n ×n ×n assembly of elements to fix ideas), we do not have one spurious pressure mode but 3n − 2 of them. This will also mean the same number of compatibility conditions on data so that trouble should be expected when using apparently reasonable boundary conditions. These spurious modes are depicted in the following Fig. 8.21. One of them is the genuine 3-D chequerboard mode (Fig. 8.21a). The other ones are built from an assembly of 2-D nodes. In Fig. 8.21b, we have sliced the mesh in order to make apparent the internal structure of this mode. There are 3(n − 1) possible such slices so that we find the number of modes stated above.
Pressure modes
We now sketch the extension of the above proof to the 3-D case. We shall only present the rectangular case to avoid lengthening unduly this exposition. We thus suppose that the mesh is built from 2 ×2 ×2 macro-elements (Fig. 8.22).
Degrees of freedom for \(\hat{V }_{h}\)
Our pressure space \(\hat{Q}_{h}\) will be built from Q h by deleting on each macro-element four = (3 ×2 − 2) spurious modes sketched in Fig. 8.23.
Local spurious modes
The mode depicted in Fig. 8.23b has obviously two other symmetrical counterparts. On each macro-element, we thus keep the 3-D analogues of the basis functions ϕ 1, M , ϕ 2, M , ϕ 3, M of Fig. 8.18. We must now introduce \(\hat{V }_{h}\). This is done again by taking off some degrees of freedom from V h . The remaining ones are sketched in Fig. 8.22 The internal node at the barycentre of the element is also used. It is now clear that \((\hat{V }_{h},\;\hat{Q}_{h})\) is a stable pair that provides O(h) convergence. There remainsto check condition (5.3.19) that is that \(\hat{V }_{h}\) is transparent with respect to \(\tilde{Q}_{h}\). This is done exactly as in the 2-D case by a simple check of flow balance at the surface of elements. Proposition 5.3.1 then applies and we get O(h) convergence for velocities and filtered pressures.
It could be hoped that the same kind of analysis could be done for equal interpolation continuous pressure methods such as the \(\underline{Q}_{1} - Q_{1}\) approximation. Unfortunately, we know of no way in which condition (5.3.19) could be made to hold and an analysis of the convergence properties of these approximations remains an open question. We can however introduce an alternate way of stabilising such approximations and this is done in the following section.
Remark 8.10.2 (However, this is a dangerous element).
As we stated at the beginning of this Section, the \(\underline{Q}_{1} - P_{0}\) element is widely employed. We presented the above results to provide the reader with enough information about this unduly popular element. It remains that using an unstable element is a dangerous option and that the price to pay for an apparent simplicity may be inaccurate results. □
Remark 8.10.3 (The worst drawback).
An important draw-back for the \(\underline{Q}_{1} - P_{0}\) element is that the condition number of the dual problem in p is mesh dependent while it is not for stable Stokes elements. When an iterative solution method is used, this leads to a strong slowdown of the convergence. This is especially disastrous for 3-D problems where iterative methods are likely to be necessary. □
8.11 Eigenvalue Problems
We shall briefly consider, here, the application of the results of Chap. 6 to the approximation of eigenvalues for the Stokes problem. The results will also be applicable to incompressible elasticity. They have some importance in this case because of the popular modal method in which a problem is approximated using a few eigenvectors as a basis for a Galerkin’s method.
We thus consider the eigenvalue problem introduced in (1.3.84), which we recall for simplicity. We now take \(V = {(H_{0}^{1}(\Omega ))}^{2}\) and \(Q = {L}^{2}(\Omega )\big/\mathbb{R}\) and we look for \(\underline{u} \in V\) and q ∈ Q satisfying
The Lagrange multiplier p ensures the incompressibility of the eigenmodes. This is a problem of the type \(\left (f,\,0\right )\). It is easy to see, in the notation of Sect. 1.2.1, that if Ω is, for instance, a convex polygon, Q 0 H is \({H}^{1}(\Omega )\big/\mathbb{R}\) and \(V _{0}^{H}\) is the subspace of \({({H}^{2}(\Omega ) \cap H_{0}^{1}(\Omega ))}^{2}\) made of divergence-free functions (see [267]). In particular, we can check that \(\vert \vert \underline{u}\vert \vert _{V _{0}^{H}} = \vert \vert \Delta \underline{u}\vert \vert _{0} \sim \vert \vert \underline{u}\vert \vert _{2}\) and \(\vert \vert p\vert \vert _{Q_{0}^{H}} = \vert \vert \mathop{\mathrm{g\underline{ra}d}}\nolimits p\vert \vert _{0}\).
Let V h and Q h be finite dimensional subspaces of V and Q respectively. We then consider the discrete version of (8.11.1),
In order to apply the theory of Sect. 1.2.1, we must check a few conditions. With respect to ellipticity, we have no problem with conforming approximations. The weak approximability (6.5.41) of \(Q_{0}^{H}\) will surely hold if
which is satisfied by all choices of finite element spaces that one may seriously think to use in practice.
The strong approximability (6.5.42) of \(V _{0}^{H}\), which now reads
is more delicate as \(\underline{{u}}^{I}\) has to be chosen in KerB h . If the pair \((V _{h},Q_{h})\) satisfies the inf-sup condition, then the property trivially holds.
Remark, however, that the typical way of proving the inf-sup condition, using a B-compatible operator (Sect. 8.4.1) for every \(\underline{u}\), is more difficult than proving (8.11.3) directly. Moreover, there are choices of elements that fail to satisfy the inf-sup condition, for which (8.11.3) holds true. For instance, we may think of the \(\underline{Q}_{1} - P_{0}\) element of Sect. 8.10.2.
Let us assume, for simplicity, that Ω is a square and that the decomposition \(\mathcal{T}_{h}\) is made by N ×N macro-elements M as in Fig. 8.17. We have seen that this choice of elements does not satisfy the inf-sup condition: the operator \(B_{h}^{t}\) has a non-trivial kernel (the chequerboard mode), and by discarding it, we still have at best a discrete inf-sup condition with β h ∼ h (see [101, 264, 314]). Nevertheless, for \(u \in V _{0}^{H} \subset \mathrm{Ker}B\), we can construct \(\underline{{u}}^{I}\) as in the construction of \(\hat{V }_{h}\) in Sect. 8.10.2: let \(\hat{\underline{u}}_{h}\) be the vector in \(\hat{V }_{h}\) satisfying (8.10.12)–(8.10.14). It is not difficult to check that \(\underline{{u}}^{I} =\hat{\underline{ u}}_{h}\) satisfies (8.11.3) with ω 2(h) = O(h). We have here an example where the eigenvalues are approximated correctly even though the global matrix associated to (6.5.8) is singular. The same kind of construction could be extended to a mesh of macro-elements as in Fig. 8.19.
Remark 8.11.1.
We have thus another instance in which the \(\underline{Q}_{1} - P_{0}\) element, very popular for incompressible elasticity problems, manages to give an impression of rectitude. □
8.12 Nearly Incompressible Elasticity, Reduced Integration Methods and Relation with Penalty Methods
8.12.1 Variational Formulations and Admissible Discretisations
We have already seen in Chap. 1 and in Remark 8.1.2 that there are difficulties associated to approximations of nearly incompressible materials when using the standard variational principle. This section will be devoted to showing how these problems arise and how they can be cured from a proper mixed formulation. Consider, to make things simpler, a problem with homogeneous Dirichlet conditions,
We already noted in Sect. 8.1 that this problem is closely related to the penalty method used to solve the Stokes problem.
It was soon recognised in practice that a brute force use of (8.12.1) could lead, for large values of λ, to bad results, the limiting case being the locking phenomenon that is an identically zero solution.
Example 8.12.1.
The simplest case of such a bad situation would be to employ piecewise linear elements. Then, for λ large, (8.12.1) forces the piecewise constant divergence to be almost null on each element, that is, implicitly using the \(\underline{P}_{1} - P_{0}\) element of Example 8.3.2. This led to the still persistent idea that triangular or tetrahedral meshes could not be used for elasticity problems. □
A cure was found in using a reduced (that is, inexact) numerical quadrature when evaluating the term \(\lambda \int _{\Omega }\vert \mathop{ \mathrm{div}}\nolimits \underline{{v}\vert }^{2}\;\mathit{dx}\) associated with compressibility effects. We refer the reader to the papers of [287] and [60] for a discussion of the long history of this idea. We shall rather develop in detail in this example the relations of reduced integrations and mixed methods and try to make clear to what extent they may be claimed to be equivalent. For this, we first recall from Chap. 1 that problem (8.12.1) can be transformed by a straightforward application of duality techniques into a saddle point problem
for which optimality conditions are, denoting \((\underline{u},p)\) the saddle point,
This is obviously very close to a Stokes problem and is also an example of the perturbed problem studied in Chap. 4, that is: find u ∈ V and q ∈ Q such that
We then know from Chap. 5, Sect. 5.5.2, that an approximation of (8.12.3) and (8.12.4) (that is, a choice of an approximation for both \(\underline{u}\) and p) which leads to error estimates independent of λ must be a good approximation for the limiting case λ = 0.
Remark 8.12.1.
The preceding sections of this chapter therefore give us a good idea of what should (or should not) be used as an approximation. All stable elements of Sects. 8.6 and 8.7 can be employed and the choice depends on the choice of solver and the mesh generation algorithm. □
What we shall now see is that reduced integration methods correspond to an implicit choice of a mixed approximation with a discontinuous pressure approximation. The success of the reduced integration method will thus rely on the qualities of this underlying mixed method. We have seen in Sect. 8.8.1 that discontinuous pressure imposing exactly the divergence-free condition requires high degree polynomials and special meshes. Reduced integration is then a way of reducing the degree of the underlying pressure in order to hopefully obtain a stable approximation.
8.12.2 Reduced Integration Methods
Let us consider a (more or less) standard approximation of the original problem (8.12.1). An exact evaluation of the “penalty term” \(\lambda \int _{\Omega }\vert \text{div}\underline{{v}\vert }^{2}\mathit{dx}\) means that for λ large, one tries to get an approximation of \(\underline{u}\) which is exactly divergence-free. However, as we have already seen, few finite elements can stand such a condition that will in most cases lead to a locking phenomenon due to over-constraining. In a mixed formulation, one relaxes the incompressibility condition by the choice of the approximation for p. Let us now see how this will be translated as a reduced integration method at least in some cases. Let us then consider \(V _{h} \subset V :=\) \({(H_{0}^{1}(\Omega ))}^{n},\;Q_{h} \subset Q := L_{0}^{2}(\Omega )\), these approximation spaces being built from finite elements defined on a partition of Ω. On each element K, let there be given a set of k points x i and weights ω i defining a numerical quadrature formula
Remark 8.12.2.
It will be convenient to define the numerical quadrature on a reference element K and to evaluate integrals by a change of variables,
The presence of the Jacobian J(x) should be taken into account when discussing the precision of the quadrature rule on K. □
Let us now make the hypothesis that for \(\underline{v}_{h} \in V _{h}\) and \(p_{h},\;q_{h} \in Q_{h}\), one has exactly
and
Let us now consider the discrete form of (8.12.4),
When the space Q h is built from discontinuous functions, this can be read element by element as
so that using (8.12.9) and (8.12.10), one gets
Formula (8.12.8) can in turn be used in the discrete form of (8.12.3) which now gives
In general, the term \(\sum _{K}\left (\sum \limits _{i=1}^{k}\omega _{i}\;J(\hat{x}_{i})(\widehat{\text{div}\underline{u}_{h}}(\hat{x}_{i}))(\widehat{\text{div}\underline{v}_{h}}(\hat{x}_{i}))\right )\) is not an exact evaluation of \(\int _{\Omega }\text{div}\underline{u}_{h}\;\text{div}\;\underline{v}_{h}\;\mathit{dx}\) and reduced integration is effectively introduced. In the case where (8.12.9) and (8.12.10) hold, there is a perfect equivalence between the mixed method and the use of reduced integration. Whatever will come from one can be reduced to the other one. It will however not be possible, in general, to get equalities (8.12.9) and (8.12.10) and therefore, a further analysis will be needed. However, we shall first consider some examples of this complete equivalence case.
Example 8.12.2.
Let us consider the \(\underline{Q}_{1} - P_{0}\) approximation on a rectangle and a one-point quadrature rule. It is clear that \(\mathop{\mathrm{div}}\nolimits \underline{u}_{h} \in P_{1}(K)\) and is integrated exactly. In the same way, a one-point rule is exact for \(\int _{\Omega }p_{h}\;q_{h}\;\mathit{dx}\) whenever \(p_{h},\;q_{h} \in P_{0}(K)\). There is thus a perfect equivalence between reduced integration and the exact penalty method defined by (8.12.11). □
Example 8.12.3.
We now consider again the same \(\underline{Q}_{1} - P_{0}\) element on a general quadrilateral. To show that we still have equivalence requires a somewhat more delicate analysis. Indeed, at first sight, the quadrature rule is not exact for \(\int _{\hat{K}}\widehat{\text{div}\underline{u}_{h}}\;J_{K}(\hat{x})\;\mathit{d\hat{x}}\). Let us however consider in detail the term \(\widehat{\text{div}\underline{u}_{h}} = \frac{\widehat{\partial u_{1}}} {\partial x_{1}} + \frac{\widehat{\partial u_{2}}} {\partial x_{2}}\). Let B = D F be the Jacobian matrix of the transformation F from \(\hat{K}\) into K. Writing explicitly
one has
so that we get
However,
When computing \(\int _{\hat{K}}\widehat{\mathop{ \mathrm{div}}\nolimits \underline{u}_{h}}\;J(\hat{x})\;\mathit{d\hat{x}}\), the Jacobians cancel and one is left with the integral of a function which is linear in each variable and which can be computed exactly by a one-point formula. □
Example 8.12.4.
Using a 4-point integration formula on a straight-sided quadrilateral can be seen, as in the previous example, to be exactly equivalent to a \(\underline{Q}_{2} - Q_{1}\) approximation [59, 60]. □
The above equivalence is, however, not the general rule. Consider the following examples.
Example 8.12.5.
We want to use a reduced integration procedure to emulate the Crouzeix-Raviart element (cf. Sect. 8.6.2). To define a P 1 pressure, we need three integration points which can generate a formula that will be exact for second degree polynomials (but not more). The bubble function included in velocity, however, makes that \(\text{div}\underline{u}_{h} \in P_{2}(K)\) and \(\int _{K}\text{div}\underline{u}_{h}\;q_{h}\mathit{dx}\) will not be evaluated exactly. □
Example 8.12.6.
A full isoparametric \(\underline{Q}_{2} - Q_{1}\) element is not equivalent to its four-point reduced integration analogue. □
Example 8.12.7.
A \(\underline{Q}_{2} - P_{0}\) approximation is not, even on rectangles, equivalent to a one-point reduced integration method, for \(\text{div}\underline{u}_{h}\) contains second order terms which are not taken into account by a one-point quadrature. □
8.12.3 Effects of Inexact Integration
If we now consider into more details the cases where a perfect equivalence does not hold between the mixed method and some reduced integration procedure, we find ourselves in the setting of Sect. 5.5.4. In particular, \(b(\underline{v}_{h},q_{h})\) is replaced by an approximate bilinear form \(b_{h}(\underline{v}_{h},q_{h})\). We shall suppose, to simplify, that the scalar product on Q h is exactly evaluated. Two questions must then be answered.
-
Does b h (. , . ) satisfy the inf-sup condition?
-
Do error estimates still hold without loss of accuracy?
We have already introduced in Sect. 5.5.4 a general setting in which this situation can be analysed. We shall first apply Proposition 5.5.8 in order to check the inf-sup condition for two examples and we shall give an example where an inexact integral changes the nature of the problem. We shall then consider consistency error for those three examples.
Example 8.12.8.
We in fact come back to Example 8.12.7 and study on a rectangular mesh the \(\underline{Q}_{2} - P_{0}\) approximation (see Sect. 8.6.3) with a one-point quadrature rule. This is not, as we have said, equivalent to the standard \(\underline{Q}_{2} - P_{0}\) approximation. We now want to check, using Proposition 8.4.1, that it satisfies the inf-sup condition. We thus have to build a continuous operator (in H 1(Ω)-norm) such that
where M 0, K is the barycentre of K and q K the restriction of q h to K. As q h is discontinuous, we can restrict our analysis to one element and we study both sides of equality (8.12.20). We have of course, taking q K = 1,
Numbering for the Q 2 element
Using the numbering of Fig. 8.24 and denoting by u i , v i the horizontal and vertical components of velocity at node i, we can write (8.12.21) by Simpson’s quadrature rule in the form
If we write
where \(\hat{u}_{4},\hat{u}_{6},\hat{v}_{6}\) and \(\hat{v}_{2}\) are corrections with respect to a bilinear interpolation, we may rewrite (8.12.22) as
On the other hand, area \((K)\;\text{div}\underline{u}_{h}(M_{0,K})\) can be seen to be equal to
If we thus split \(\underline{u}_{h}\) into a bilinear part \(\underline{u}_{h}^{0}\) and a mid-point correction part \(\hat{\underline{u}}_{h}\), one can define \(\Pi _{h}\underline{u}_{h}\) by setting
Equality (8.12.21) will then hold and (8.12.27) is clearly continuous with a continuity constant independent of h. □
Example 8.12.9.
We come back to Example 8.12.5 that is a three-point quadrature rule used in conjunction with the Crouzeix-Raviart element. We shall not give the analysis in detail but only sketch the ideas. The problem is again to check that the inf-sup condition holds through Proposition 8.4.1. As the quadrature rule is exact when q h is piecewise constant, the obvious idea is to build \(\Pi _{h}\underline{u}_{h}\) by leaving invariant the trace of \(\underline{u}_{h}\) on ∂ K and only modifying the coefficients of the bubble functions. This can clearly be done. Continuity is now to be checked and the proof is essentially the same as the standard proof of the inf-sup condition (Sect. 8.7.2). □
Example 8.12.10 (A modified \(\underline{Q}_{1} - P_{0}\) element).
We now present a puzzling example [127] of an element which is stable but for which convergence is tricky due to a consistency error term. We have here a case where using a one-point quadrature rule will change the situation with respect to the inf-sup condition. In fact, it will make a stable element from an unstable one but will also introduce an essential change in the problem. The departure point is thus the standard Q 1 − P 0 element which was studied in Sect. 8.10.2 and which, as we know, does not satisfy the inf-sup condition. We now make it richer by adding to velocity \(\underline{u}_{h}\vert _{K} =\{ u_{1},u_{2}\}\) what we shall call wave functions. On the reference element \(\hat{K} =] - 1,1[\times ] - 1,1[\), those functions are defined by
where \(b_{2}(\hat{x},\hat{y}) = (1 -\hat{ {x}}^{2})(1 -\hat{ {y}}^{2})\) is the Q 2 bubble function. If we now consider
we obtain a new element with an internal degree of freedom. The wave functions that we added vanish on the boundary and nothing is changed for the stability of the mixed method with exact integration. If we rather use a one-point quadrature rule, things become different. We shall indeed check that the modified bilinear form \(b_{h}(\hat{\underline{v}}_{h},q_{h})\) satisfies the inf-sup condition. We thus have to show that
This is easily checked by posing on K (we suppose that we have a rectangular mesh to make things simpler)
We then have \(\text{div}\hat{\underline{u}}_{h} = p_{h}\) and
which implies
and (8.12.30) follows. A remarkable point is, now, that even the hydrostatic mode has disappeared. This is an indication that something incorrect has been introduced in the approximation. An analysis of consistency error indeed shows that usual error estimates fail and that we are actually approximating a continuous problem in which the incompressibility condition has been replaced by \(\text{div}\underline{u} + kp = 0\) where \(k = a/b\). We then see that if, in general for the Stokes problem, making the space of velocities richer improves (at least does not reduce) the quality of the method, this fact can become false when numerical integration is used. □
Let us now turn our attention to the problem of error estimation. From Proposition 5.5.6 and Remark 5.5.9, all we have to do is to estimate the consistency terms,
and
We thus have to estimate quadrature errors. It would be out of purpose to enter into details, and we refer the reader to [147, 148] where examples of such analysis are presented exhaustively. The first step is to transform (8.12.34) into a form which is sometimes more tractable. We may indeed write
and
The first parenthesis in the right-hand side of (8.12.36) and (8.12.37) can be reduced to an approximation error. The second parenthesis implies only polynomials.
Let us therefore consider (8.12.37) for the three approximations introduced above. For the Crouzeix-Raviart triangle, taking \(\underline{v}_{h}\) the standard interpolate of \(\underline{u}\) makes the second parenthesis vanish while the first yields an O(h) estimate. For the two other approximations, taking \(\underline{v}_{h}\) to be a standard bilinear approximation of \(\underline{u}\) makes the second parenthesis vanish while the first yields an O(h) estimate, which is the best that we can hope for anyway. The real trouble is therefore with (8.12.34), with or without (8.12.36). In the case of the Crouzeix-Raviart triangle, we can use directly (8.12.34) and the following result of [147, 148].
Proposition 8.12.1.
Let \(f \in W_{k,q}(\Omega ),\;p_{k} \in P_{k}(K)\) and denote E k (fp k ) the quadrature error on element K when numerical integration is applied to fp k . Let us suppose that \(E_{K}(\hat{\phi }) = 0\;\forall \,\hat{\phi }\in P_{2k-2}(K)\) . Then, one has
□
Taking \(k = 2\;,q = \infty \) and using the inverse inequality to go from \(\vert p_{k}\vert _{1}\) to \(\vert p_{k}\vert _{0}\), one gets an O(h 2) estimate for (8.12.34).
The two other approximations cannot be reduced to Proposition 8.12.1 and must be studied through (8.12.36). We must study a term like
This can at best be bounded. For instance in the case of the \(\underline{Q}_{2} - P_{0}\) approximation, we can check by hand that the quadrature error on K reduces to \(h_{K}^{3}\;\vert \text{div}\underline{v}_{h}\vert _{2,K}\,p_{k}\).
8.13 Other Stabilisation Procedures
We shall now consider, for the Stokes problem, stabilised formulations presented in Sect. 6.1.1 of Chap. 6. It is clear or should be clear from the results presented in the present chapter that the key of success in stabilising incompressible elements is in weakening the discrete divergence-free condition. This was done, up to now, by reducing the space Q h of pressures or by enriching the space V h of velocity field. We now consider the other possibility of a modified variational formulation. In many cases, this will amount to explicitly weaken the condition \(\text{div}_{h}\;\underline{u}_{h} = 0\) by changing it to
where g h is a (well chosen) “small” function. One step in this direction had been done in the work of [131] who considered the relaxed condition
in the case of a continuous pressure approximation (that is \(Q_{h} \subset {H}^{1}(\Omega )\)). In (8.13.2), β is any positive real number. On a regular mesh, this is a discrete form of
It is easy to understand that appearance of oscillations due to spurious pressure modes will make \(\bigtriangleup _{h}\;p_{h}\) large. This will relax the divergence-free condition, thus preventing the growth of such oscillations. The practical use of (8.13.2) indeed requires a delicate balance between two conflicting phenomena. When α is chosen too small, stabilisation is poor and spurious pressure modes persist. On the other hand, taking α too large spoils the value of p h near the boundary because of the parasitic Neumann condition \(\frac{\partial p_{h}} {\partial n} = 0\) which is implicit in (8.13.2).
This procedure was later generalised by Hughes and Franca [256] and Hughes et al. [257] in order to improve its consistency, in a way that we present below.
We shall first try to give a unified presentation of this kind of methods using the general theory of stabilisation procedures developed in Chap. 5. We shall first consider augmented methods.
8.13.1 Augmented Method for the Stokes Problem
We consider the stabilised formulation (6.1.50) in the special context of the Stokes problem (1.5.23). We now have
and the dense inclusions V ⊂ H ⊂ V ′. We also have, for \(\underline{u} \in V\) and p ∈ Q,
As to the operator A, it is defined by
The case t = 1 of Sect. 6.1.2 (with β 2 = 0) which corresponds to a stabilisation method introduced by Douglas and Wang [179] now reads as
while in the case t = 0, the extra term is only written in the second equation
We must now build a computable implementation of these formulations. Indeed, the scalar product in V ′ is not directly handable. In the present case, as the operator A is an isomorphism from V onto V ′, we can define the scalar product as
However, this means being able to compute the exact inverse of A. We now have to introduce an approximation and many options are open.
Example 8.13.1 (Defining a scalar product by an approximation of A −1).
The first idea that comes to mind is to use some approximate operator \(S_{h}^{-1}\) instead of A − 1. This could be done by solving an auxiliary problem in a space richer than V h . Our problem (8.13.7) would now be changed into
with a similar expression for (8.13.6). □
Example 8.13.2 (Defining a scalar product by a change of space).
Another way of defining a discrete formulation, introduced in [256] and [257], is to replace
by an expression of the form
and by a similar change on the term appearing in the first equation of (8.13.6) if we want to use t = 1. This can be seen as another way (through an “inverse inequality”) of defining a discrete scalar product corresponding to the H − 1(Ω) scalar product by using a scalar product in H. If the exact solution \(\underline{u}\) is regular enough, this expression written for \(\underline{u}\) vanishes and we have what has been termed as “strong consistency”. Note that for piecewise linear \(\underline{u}_{h},\;A\underline{u}_{h}\vert _{K}\) vanishes, leaving only a corrective term of the form
which is a variant of (8.13.2). The aim of (8.13.11) is largely to replace the Neumann condition \(\frac{\partial p_{h}} {\partial n} = 0\), which is implicit in (8.13.2), by a more correct one, hopefully making the choice of β easier. □
We now come back to our discrete stabilised problems (8.13.6) or (8.13.9) and we first consider the option of using a discrete operator \(S_{h}^{-1}\). We shall see that in one important case, both options are equivalent, and that in other cases, we fall back onto known methods.
8.13.2 Defining an Approximate Inverse \(S_{h}^{-1}\)
Let V h be the finite element space in which we compute \(\underline{u}_{h}\). We introduce a space \(V _{h}^{+}\) of new degrees of freedom and the space \(W_{h} := V _{h} \oplus V _{h}^{+}\). We can now define \(\underline{s}_{h}^{+} := {S}^{-1}(A\underline{u}_{h} +\mathop{ \mathrm{g\underline{ra}d}}\nolimits p_{h} -\underline{ f}) \in V _{h}^{+}\) by a “hierarchical” computation in \(V _{h}^{+}\)
In many cases, \(V _{h}^{+}\) will be a space of “bubbles” and this problem will be solvable element by element. For the case t = 0, our stabilised problem (8.13.7) would now be read, \(\underline{s}_{h}^{+}\) being defined from (8.13.13), as
which indeed contains a weakened condition of the form (8.13.1). The case t = 1 of (8.13.6) would yield
Taking into account equation (8.13.13), one can see that for β = 1, this is nothing but the solution of the Stokes problem using W h ×Q h as the finite element space. This will obviously work if this choice of spaces is stable and we have rediscovered that enriching the space V h is a good way of getting a stable method. As for the case (8.13.14), it would be an approximation of this case, which could hardly be considered as simpler as the matrix of the problem is not symmetric. Proving its stability would require some “quasi-orthogonality” between V h and \(V _{h}^{+}\) of the same type as what is used to study hierarchical error estimators [2, 47]. We shall not try to get much further in this direction. There is however a simple case where both (8.13.14) and (8.13.15) coincide and in fact are equivalent to (8.13.12).
Example 8.13.3 (Defining \(\mathbf{S_{h}^{-1}}\) with bubbles, the MINI element).
Let us consider the case of a piecewise linear approximation for both V h and Q h , which is well known to be unstable. To fix ideas, we shall use for \(V _{h}^{+}\) the space B 3 of conforming cubic bubbles, although we might also use other shapes of bubbles or nonconforming quadratic bubbles. Let then b K be the bubble associated to element K. We can write any function of \(V _{h}^{+}\) in the form
The key of what follows is the fact that we have orthogonality between the space of bubbles and V h in the sense that
As we use bubbles, our Eq. (8.13.13) can be solved element by element and we have on every K
where
To make things easier, suppose that f is piecewise constant so that we can rewrite (8.13.17) as
where we denote
We thus obtain
Using (8.13.21), (8.13.14) becomes
and this is nothing but a slight variant of the stabilisation obtained from (8.13.11), as we can check that
with a constant c K depending on the shape of the element. The same reasoning can be done with other choices for the space of bubbles.
Remark 8.13.1 (The MINI element).
Given the orthogonality of (8.13.16), it is easy to see that the formulations of (8.13.6) and (8.13.7) obtained from (8.13.17) coincide and that for β = 1, they are nothing but the solution of the Stokes problem with the MINI element. □
We thus see that the technique of Example 8.13.2 can be obtained in different ways. We now proceed to develop an error analysis of these methods.
Example 8.13.4 (Error estimates for the Hughes-Franca stabilisation).
We place ourselves in the case of “equal interpolation”, that is, using polynomials in \(\mathcal{L}_{k}^{1}\) for V h and Q h . Note however that the space V h will satisfy boundary conditions while Q h will not. We present the result in the two-dimensional case but it can easily be extended to the three-dimensional case. We have a space of continuous pressures and we have thus \(\mathop{\mathrm{g\underline{ra}d}}\nolimits p_{h} \in H = {({L}^{2}(\Omega ))}^{2}\). To simplify the presentation, we define on H
For any \(\underline{u}_{h} \in V _{h}\), we also define \(\underline{A}\underline{u}_{h} \in H\) by
and we write a stabilised formulation
which is the method of [256].
Following the procedure of Chap. 5, Sect. 6.1.2, we must obtain a stability result and estimate the consistency term. For stability, we define the bilinear form
For \(\underline{u}_{h} =\underline{ v}_{h}\) and p h = q h , we obtain
To prove stability, we recall that from Verfürth’s trick (cf. Sect. 8.5.2 and also Chap. 6.3), we have
On the other hand, we bound the last term by
However, using an inverse inequality, we have
and thus
Using this last result in (8.13.28), we have
which implies stability for β small enough. It should be remarked that for the degree of the approximation k = 1, \(A\underline{u}_{h}\vert _{K} = 0\) and that we then have stability for any value of β.
Following Sect. 6.1.1 of Chap. 6, we now have to bound, \((\underline{u}_{I},p_{I})\) being an interpolate of \((\underline{u},p)\), a term of the form
The crucial term is the first one. If polynomials of degree k are employed, we have, by classical interpolation results, an estimate on K,
and the loss of precision is exactly compensated by the choice of the stabilising parameter \(\beta h_{K}^{2}\).
The method does work for any degree of polynomial. However, it should be noted that for k > 1, it leads to a non-symmetric system which makes it less appealing. □
8.13.3 Minimal Stabilisations for Stokes
We now consider another class of stabilisations which contains, as a special case, the method of (8.13.2). Although this could be written in general, we shall restrict ourselves to the case of a first order approximation, \(V _{h} \subset {(\mathcal{L}_{1}^{1})}^{n} \cap V,Q_{h} \subset \mathcal{L}_{k}^{1} \cap Q\). The method will be a direct adaptation of Sect. 6.3.1 of Chap. 6. We introduce another space \(\tilde{V }_{h} \in {(\mathcal{L}_{1}^{1})}^{2}\), denoting \(\tilde{P}\) the projection over \(\tilde{V }_{h}\) in the norm of \(H := {({L}^{2}(\Omega ))}^{n}\) and we consider the following problem:
This fits entirely into the theory of Chap. 6 and we have the error bound
provided that the following assumption holds:
Now, as we use an approximation of degree one, we would like all terms on the right-hand side of inequality (8.13.36) to be O(h 2) and we consider three cases.
-
(i)
The Brezzi-Pitkäranta formulation. If we take \(\tilde{V }_{h} =\{ 0\}\) and thus \(\tilde{P} = I\), assumption A evidently holds. The last term in (8.13.36) reduces to \(r\|\mathop{\mathrm{g\underline{ra}d}}\nolimits p\|_{H}^{2}\) and we need to make r = O(h 2) to get the error bound. The drawback of the method is the boundary layer on \(\mathop{\mathrm{g\underline{ra}d}}\nolimits p_{h}\).
-
(ii)
Projection on V h. Assumption A is again immediate. We must again consider the last consistency term. Some trouble arises because V h satisfies boundary conditions (e.g., \(\underline{v}_{h} = 0\) on the boundary). We can then expect \(\|(I -\tilde{ P})(\mathop{\mathrm{g\underline{ra}d}}\nolimits p)\|_{H}^{2}\) to be no better than O(h). Taking r = O(h) would restore the optimal order but the problem with the boundary layer is not cured.
-
(iii)
Optimal projection. From the previous discussion, one sees that taking \(\tilde{V }_{h} = {(\mathcal{L}_{1}^{1})}^{2}\), that is, suppressing boundary conditions, will work with r = O(1) and will eliminate the spurious boundary layer. The trouble is now with assumption A. It was proved in [123] that it indeed holds. This method had been used in [51].
In order to prove our assumption A, we first consider the following result.
Proposition 8.13.1.
Let Q h and V h be the space of piecewise linear pressures and velocities as above, and let \(\tilde{V }_{h}\) be the space of piecewise linear continuous vectors on \(\mathcal{T}_{h}\) (without boundary conditions.) There exists a constant β ∗ > 0, independent of h, such that, for every q h ∈ Q h and for every \(\underline{w}_{h} \in \tilde{ V }_{h}\) , there exists a \(\underline{v}_{h}^{0} \in V _{h}\) verifying
and
Proof.
Let us consider first a macro-element K made by the collection of triangles having one vertex P of \(\mathcal{T}_{h}\) in common. Split \(q_{h} = q_{0} + q_{\ell}\), where q 0 is such that \(\mathop{\mathrm{g\underline{ra}d}}\nolimits q_{0}\) has zero mean value in K and q ℓ is linear on K (hence \(\mathop{\mathrm{g\underline{ra}d}}\nolimits q_{\ell}\) = constant in K.) It is clear that \((\mathop{\mathrm{g\underline{ra}d}}\nolimits q_{0},\mathop{\mathrm{g\underline{ra}d}}\nolimits q_{\ell})_{K} = 0\). We now take \(\underline{v}_{h}^{0}\), piecewise linear, continuous, vanishing on the boundary of K and having value \(\sqrt{ 6}\mathop{\mathrm{g\underline{ra}d}}\nolimits q_{\ell}\) at the internal vertex P. An easy computation shows that:
and
On the other hand, \(\mathop{\mathrm{g\underline{ra}d}}\nolimits q_{0}\) belongs to a space (piecewise constant vectors on K, with continuous tangential components, and zero mean on K) whose intersection with piecewise linear continuous vectors on K is reduced to the zero vector. As we are in finite dimension, there exists a positive constant δ K such that, for every \(\mathop{\mathrm{g\underline{ra}d}}\nolimits q_{0}\) and for every \(\underline{w}_{h}\),
As \(\mathop{\mathrm{g\underline{ra}d}}\nolimits q_{\ell}\) is clearly continuous and piecewise linear, (8.13.42) easily implies that
and a simple scaling argument shows immediately that δ K is independent of the size of K (notice that (8.13.43) holds for every \(\underline{w}_{h}\)).
Finally, we explicitly point out that
where P is the only vertex internal to K. From (8.13.41) to (8.13.44), one then gets that, for every q h and for every \(\underline{w}_{h}\), there is a \(\underline{v}_{h}^{0}\), piecewise linear, continuous, and vanishing on the boundary of K, such that (8.13.40) holds and
for some positive constant β K independent of q h and \(\underline{w}_{h}\). The result (8.13.38) and (8.13.39) then follows easily from (8.13.45) by typical instruments (continuity of β K , splitting of Ω into macro-elements such that each triangle belongs at most to three different macro-elements, and so on). □
With the aid of Proposition 8.13.1, we can now prove Assumption A.
Proposition 8.13.2.
Let Q h , V h and \(\tilde{V }_{h}\) be as in Proposition 8.13.1 . Then, there exists a constant \(\tilde{\beta }> 0\) such that
where all the norms are in L 2 .
Proof.
We start by observing that, for every \(\underline{v}_{h}^{0}\) and q h , we have
where the last inequality clearly holds for every positive β ∗ , but we shall use it for the value of β ∗ given in (8.13.39). For every q h , we now take \(\underline{v}_{h}^{0}\) as given by Proposition 8.13.1, and using (8.13.38), we have
which, inserted in (8.13.39) with \(\underline{w}_{h} = P_{\tilde{V }_{h}}\mathop{ \mathrm{g\underline{ra}d}}\nolimits q_{h}\), gives
and (8.13.46) follows immediately. □
Remark 8.13.2 (Enhanced strain methods).
Finally, to conclude this section, we would like to note that another example of stabilisation of the Stokes problem by an enhanced method can be found in the work of [283]. □
8.14 Concluding Remarks: Choice of Elements
We would first like to emphasise that the results of this chapter can be applied as well to flow problems as well as to linear (or linearised) elasticity problems. In this last case, displacement methods also need to be considered from a mixed point of view. Indeed, we have already seen in Sect. 8.12 that there is a close relation between the Stokes problem and linear elasticity problems. However, things are not so simple: fluid people and solid mechanics people form two different communities and information was long to cross the border.
8.14.1 Choice of Elements
We have presented discontinuous pressure and continuous pressure elements. They both have advantages, even though discontinuous pressure is appealing as it enforces an element wise conservation of mass. They can also be implemented by a penalty procedure.
In this respect, the reader should have noticed an important difference between the two-dimensional and the three-dimensional elements presented in this chapter.
-
In the 2-D case, we have a choice of discontinuous pressure elements which can be used with a penalty method. Direct solvers are not too sensitive to the ill-conditioning of the resulting system and we thus obtain a good resolution strategy. We can thus recommend the Crouzeix-Raviart element of Example 8.6.1 or its higher order variants.
-
In the 3-D case, discontinuous pressure elements satisfying the inf-sup condition are expensive as one needs degrees of freedom on the faces. The equivalent of the Crouzeix-Raviart element also uses bubbles of degree four. The ubiquitous \(\underline{Q}_{1} - P_{0}\), for which the condition number of the dual problem depends on h, behaves badly with iterative methods which are essential for large-scale simulations. We therefore recommend the Hood-Taylor continuous pressure element.
Remark 8.14.1 (Solvers).
The choice of elements is also dictated by the choice of solvers. In the two-dimensional case, where direct solvers are almost always employed, discontinuous pressure elements are desirable as they are compatible with penalty methods. In the three-dimensional case, where iterative solvers are the rule for large problems, penalty methods are to be avoided as they destroy the condition number of the problem. Recent progress in the construction of solvers for indefinite systems [58, 185, 186] however make the use of a continuous pressure element, such as the Taylor-Hood element, possible and efficient. □
Remark 8.14.2 (Meshes).
Another consideration is the choice of affine (triangular or tetrahedral) or quadrilateral hexahedral elements. As we already noted, there was a widespread legend that tetrahedra were not suitable for incompressible solid mechanics problems. This was based on a lack of analysis and we advocate the choice of affine elements for two reasons.
-
Mesh generation is much easier with tetrahedra than with hexahedra. Indeed, it can, most of times, be done automatically. This is important in complex engineering problems where the domains may be of a complex shape.
-
The second reason is mesh adaptation, which is also much easier for tetrahedra. There exist algorithms which can make a mesh optimal to represent a given solution. □
Finally, let us recall that the approximation of incompressible materials is a central issue in many industrial applications. It has therefore been the subject of a vast literature. We believe to have presented the essential points but we also neglected many aspects. Among those, we did not describe finite volume methods, which are mostly amenable to an analysis by the theory of mixed methods. One can find references in [194] and [195].
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Boffi, D., Brezzi, F., Fortin, M. (2013). Incompressible Materials and Flow Problems. In: Mixed Finite Element Methods and Applications. Springer Series in Computational Mathematics, vol 44. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-36519-5_8
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