Abstract
A new weak Galerkin (WG) finite element method for solving the biharmonic equation in two or three dimensional spaces by using polynomials of reduced order is introduced and analyzed. The WG method is on the use of weak functions and their weak derivatives defined as distributions. Weak functions and weak derivatives can be approximated by polynomials with various degrees. Different combination of polynomial spaces leads to different WG finite element methods, which makes WG methods highly flexible and efficient in practical computation. This paper explores the possibility of optimal combination of polynomial spaces that minimize the number of unknowns in the numerical scheme, yet without compromising the accuracy of the numerical approximation. Error estimates of optimal order are established for the corresponding WG approximations in both a discrete \(H^2\) norm and the standard \(L^2\) norm. In addition, the paper also presents some numerical experiments to demonstrate the power of the WG method. The numerical results show a great promise of the robustness, reliability, flexibility and accuracy of the WG method.
Similar content being viewed by others
Avoid common mistakes on your manuscript.
1 Introduction
This paper will concern with approximating the solution \(u\) of the biharmonic equation
with clamped boundary conditions
where \(\Delta \) is the Laplacian operator, \(\Omega \) is a bounded polygonal or polyhedral domain in \(\mathbb {R}^d\) for \(d=2, 3\) and \(\mathbf n \) denotes the outward unit normal vector along \(\partial \Omega \). We assume that \(f, g, \phi \) are given, sufficiently smooth functions.
This problem mainly arises in fluid dynamics where the stream functions \(u\) of incompressible flows are sought and elasticity theory, in which the deflection of a thin plate of the clamped plate bending problem is sought [26, 34, 36].
Due to the significance of the biharmonic problem, a large number of methods for discretizing (1.1)–(1.3) have been proposed. These methods include dealing with the biharmonic operator directly, such as discretizing (1.1)–(1.3) on a uniform grid using a 13-point or 25-point direct approximation of the fourth order differential operator [9, 24]; mixed methods, that is, splitting the biharmonic equation into two coupled Poisson equations [1, 4–7, 12, 15, 17–20, 25, 27]. Also there are some other approaches to the biharmonic problems, like the conformal mapping methods [11, 35], integral equations [29], orthogonal spline collocation method [8] and the fast multipole methods [23], etc.
Among these methods, finite element methods are one of the most widely used technique, which is based on variational formulations of the equations considered. In fact, the biharmonic equation is also one of the most important applicable problems of the finite element methods, cf. [2, 13, 14, 16, 22, 41]. The Galerkin methods, discretizing the corresponding variational form of (1.1) is given by seeking \(u\in H^2(\Omega )\) satisfying
such that
where \(H_0^2(\Omega )\) is the subspace of \(H^2(\Omega )\) consisting of functions with vanishing value and normal derivative on \(\partial \Omega \).
Standard finite element methods for solving (1.1)–(1.3) based on the variational form (1.4) with conforming finite element require rather sophisticated finite elements such as the 21-degrees-of-freedom of Argyris (see [3]) or nonconforming elements of Hermite type. Since the complexity in the construction for the finite element with high continuous elements, \(H^2\) conforming element are seldom used in practice for the biharmonic problem. To avoid using of \(C^1\)-elements, besides the mixed methods, an alternative approach, nonconforming and discontinuous Galerkin finite element methods have been developed for solving the biharmonic equation over the last several decades. Morley element [28] is a well known nonconforming element for the biharmonic equation for its simplicity. A \(C^0\) interior penalty method was developed in [10, 21]. In [30], a hp-version interior penalty discontinuous Galerkin method was presented for the biharmonic equation.
Recently a new class of finite element methods, called weak Galerkin(WG) finite element methods were developed for the biharmonic equation for its highly flexible and robust properties. The WG method refers to a numerical scheme for partial differential equations in which differential operators are approximated by weak forms as distributions over a set of generalized functions. This thought was first proposed in [38] for a model second order elliptic problem, and this method was further developed in [31, 39, 40]. In [32], a weak Galerkin method for the biharmonic equation was derived by using discontinuous functions of piecewise polynomials on general partitions of polygons or polyhedra of arbitrary shape. After that, in order to reduce the number of unknowns, a \(C^0\) WG method [33] was proposed and analyzed. However, due to the continuity limitation, the \(C^0\) WG scheme only works for the traditional finite partitions, while not arbitrary polygonal or polyhedral girds as allowed in [32].
In order to realize the aim that reducing the unknown numbers and suit for general partitions of polygons or polyhedra of arbitrary shape at the same time, in this paper we construct a reduction WG scheme based on the use of a discrete weak Laplacian plus a new stabilization that is also parameter free. The goal of this paper is to specify all the details for the reduction WG method for the biharmonic equations and present the numerical analysis by presenting a mathematical convergence theory.
An outline of the paper is as follows. In the remainder of the introduction we shall introduce some preliminaries and notations for Sobolev spaces. In Sect. 2 is devoted to the definitions of weak functions and weak derivatives. The WG finite element schemes for the biharmonic Eqs. (1.1)–(1.3) are presented in Sect. 3. In Sect. 4, we establish an optimal order error estimates for the WG finite element approximation in an \(H^2\) equivalent discrete norm. In Sect. 5, we shall drive an error estimate for the WG finite element method in the standard \(L^2\) norm. Section 6 contains the numerical results of the WG method. The theoretical results are illustrated by these numerical examples. Finally, we present some technical estimates for quantities related to the local \(L^2\) projections into various finite element spaces and some approximation properties which are useful in the convergence analysis in “Appendix”.
Now let us define some notations. Let \(D\) be any open bounded domain with Lipschitz continuous boundary in \(\mathbb {R}^d, d=2, 3\). We use the standard definition for the Sobloev space \(H^s(D)\) and their associated inner products \((\cdot , \cdot )_{s, D}\), norms \(\Vert \cdot \Vert _{s, D}\), and seminorms \(|\cdot |_{s, D}\) for any \(s\ge 0\).
The space \(H^0(D)\) coincides with \(L^2(D)\), for which the norm and the inner product are denoted by \(\Vert \cdot \Vert _D\) and \((\cdot ,\cdot )_D\), respectively. When \(D=\Omega \), we shall drop the subscript \(D\) in the norm and in the inner product notation.
The space \(H(\mathrm{div}; D)\) is defined as the set of vector-valued functions on \(D\) which, together with their divergence, are square integrable; i.e.,
The norm in \(H(\mathrm{div}; D)\) is defined by
2 Weak Laplacain and Discrete Weak Laplacian
For the biharmonic equation (1.1), the underlying differential operator is the Laplacian \(\Delta \). Thus, we shall first introduce a weak version for the Laplacian operator defined on a class of discontinuous functions as distributions [32].
Let \(K\) be any polygonal or polyhedral domain with boundary \(\partial K\). A weak function on the region \(K\) refers to a function \(v= \{v_0, v_b, \mathbf v _g\}\) such that \(v_0\in L^2(K), v_b\in L^{2}(\partial K)\), and \(\mathbf v _g\cdot \mathbf n \in L^{2}(\partial K)\), where \(\mathbf n \) is the outward unit normal vector along \(\partial K\). Denote by \(\mathcal {W}(K)\) the space of all weak functions on \(K\), that is,
Recall that, for any \(v\in \mathcal {W}(K)\), the weak Laplacian of \(v=\{v_0, v_b, \mathbf v _g\}\) is defined as a linear functional \(\Delta _w v\) in the dual space of \(H^2(K)\) whose action on each \(\varphi \in H^2(K)\) is given by
where \((\cdot , \cdot )_K\) stands for the \(L^2\)-inner product in \(L^2(K)\) and \(\langle \cdot , \cdot \rangle _{\partial K}\) is the inner product in \(L^2(\partial K)\).
The Sobolev space \(H^2(K)\) can be embedded into the space \(\mathcal {W}(K)\) by an inclusion map \(i_\mathcal {W}: H^2(K)\rightarrow \mathcal {W}(K)\) defined as follows
With the help of the inclusion map \(i_\mathcal {W}\), the Sobolev space \(H^2(K)\) can be viewed as a subspace of \(\mathcal {W}(K)\) by identifying each \(\phi \in H^2(K)\) with \(i_\mathcal {W}(\phi )\).
Analogously, a weak function \(v= \{v_0,v_b, \mathbf v _g\}\in \mathcal {W}(K)\) is said to be in \(H^2(K)\) if it can be identified with a function \(\phi \in H^2(K)\) through the above inclusion map. Here the first components \(v_0\) can be seen as the value of \(v\) in the interior and the second component \(v_b\) represents the value of \(v\) on \(\partial K\). Denote \(\nabla v\cdot \mathbf n \) by \(v_n\), then the third component \(\mathbf v _g\) represents \((\nabla v\cdot \mathbf n )\mathbf n |_{\partial K}=v_n \mathbf n \). Obviously, \(\mathbf v _g\cdot \mathbf n = \nabla v\cdot \mathbf n \). Note that if \(v\not \in H^2(K)\), then \(v_b\) and \(\mathbf v _g\) may not necessarily be related to the trace of \(v_0\) and \((\nabla v_0\cdot \mathbf n )\mathbf n \) on \(\partial K\), respectively.
For \(v\in H^2(K)\), from integration by parts we have
Thus the weak Laplacian is identical with the strong Laplacian, i.e.,
for smooth functions in \(H^2(K)\).
For numerical implementation purpose, we define a discrete version of the weak Laplacain operator by approximating \(\Delta _w\) in polynomial subspaces of the dual of \(H^2(K)\). To this end, for any non-negative integer \(r\ge 0\), let \(P_r(K)\) be the set of polynomials on \(K\) with degree no more than \(r\).
Definition 2.1
([32]) A discrete weak Laplacian operator, denoted by \(\Delta _{w,r,K}\), is defined as the unique polynomial \(\Delta _{w, r, K} v\in P_r(K)\) satisfying
From the integration by parts, we have
Substituting the above identity into (2.3) yields
for all \(\varphi \in P_r(K)\).
3 Weak Galerkin Finite Element Scheme
Let \(\mathcal {T}_h\) be a partition of the domain \(\Omega \) into polygons in 2D or polyhedra in 3D. Assume that \(\mathcal {T}_h\) is shape regular in the sense as defined in [39]. Denote by \(\mathcal {E}_h\) the set of all edges or flat faces in \(\mathcal {T}_h\), and let \(\mathcal {E}_h^0=\mathcal {E}_h\setminus \partial \Omega \) be the set of all interior edges or flat faces.
Since \(v_n\) represents \(\nabla v\cdot \mathbf n \), then \(v_n\) is naturally dependent on \(\mathbf n \). To ensure a single valued function \(v_n\) on \(e\in \mathcal {E}_h\), we introduce a set of normal directions on \(\mathcal {E}_h\) as follows
For any given integer \(k\ge 2, T\in \mathcal {T}_h\), denote by \(\mathcal {W}_k(T)\) the discrete weak function space given by
By patching \(\mathcal {W}_k(T)\) over all the elements \(T\in \mathcal {T}_h\) through a common value on the interface \(\mathcal {E}_h^0\), we arrive at a weak finite element space \(V_h\) given by
Denote by \(V_h^0\) the subspace of \(V_h\) constituting discrete weak functions with vanishing traces; i.e.,
Denote by \(\Lambda _h\) the trace of \(V_h\) on \(\partial \Omega \) from the component \(v_b\). It is obvious that \(\Lambda _h\) consists of piecewise polynomials of degree \(k-1\). Similarly, denote by \(\Upsilon _h\) the trace of \(V_h\) from the component of \(v_n\) as piecewise polynomials of degree \(k-1\). Denote by \(\Delta _{w, k-2}\) the discrete weak Laplacian operator on the finite element space \(V_h\) computed by using (2.3) on each element \(T\) for \(k\ge 2\), that is,
For simplicity, we shall drop the subscript \(k-2\) in the notation \(\Delta _{w, k-2}\) for the discrete weak Laplacian operator. We also introduce the following notation
For each element \(T\in \mathcal {T}_h\), denote by \(Q_0\) the \(L^2\) projection onto \(P_k(T), k\ge 2\). For each edge/face \(e\subset \partial T\), denote by \(Q_b\) the \(L^2\) projection onto \(P_{k-1}(e)\). Now for any \(u\in H^2(\Omega )\), we shall combine these two projections together to define a projection into the finite element space \(V_h\) such that on the element \(T\)
Theorem 3.1
Let \(\mathbb {Q}_h\) be the local \(L^2\) projection onto \(P_{k-2}\). Then the following commutative diagram holds true on each element \(T\in {\mathcal {T}}_h\):
Proof
For any \(\phi \in P_{k-2}(T)\), from the definition of the discrete weak Laplacian and the \(L^2\) projection
which implies (3.4). \(\square \)
The commutative property (3.4) indicates that the discrete weak Laplacian of the \(L^2\) projection of \(u\) is a good approximation of the Laplacian of \(u\) in the classical sense. This is a good property of the discrete weak Laplacian in application to algorithm and analysis.
For any \(u_h=\{u_0, u_b, u_n \mathbf n _e\}\) and \(v=\{v_0, v_b, v_n \mathbf n _e \}\) in \(V_h\), we introduce a bilinear form as follows
Weak Galerkin Algorithm 1
Find \(u_h = \{u_0, u_b, u_n\mathbf n _e\}\in V_h\) satisfying \(u_b = Q_b g\) and \(u_n = Q_{b}\phi \) on \(\partial \Omega \) and the following equation:
Lemma 3.2
For any \(v \in V_h^0\), let \( |\!|\!| v |\!|\!| \) be given by
Then, \( |\!|\!| \cdot |\!|\!| \) defines a norm in the linear space \(V_h^0\).
Proof
For simplicity, we shall only prove the positivity property for \( |\!|\!| \cdot |\!|\!| \). Assume that \( |\!|\!| v |\!|\!| = 0\) for some \(v \in V_h^0\). It follows that \(\Delta _w v=0\) on each element T, \(Q_b v_0=v_b\) and \(\nabla v_0\cdot \mathbf n _e = v_n\) on each edge \(\partial T\). We claim that \(\Delta v_0 = 0\) holds true locally on each element T. To this end, for any \(\varphi \in P_{k-2}(T)\) we use \(\Delta _w v=0\) and the identify (2.4) to obtain
where we have used the fact that \(Q_b v_0-v_b=0\) and
in the last equality. The identity (3.7) implies that \(\Delta v_0=0\) holds true locally on each element \(T\).
Next, we claim that \(\nabla v_0 = 0\) also holds true locally on each element \(T\). For this purpose, for any \(\phi \in P_{k}(T)\), we utilize the Gauss formula to obtain
By letting \(\phi =v_0\) on each element \(T\) and summing over all \(T\) we obtain
For two elements \(T_1, T_2\in \mathcal {T}_h\), which share \(e\in \mathcal {E}_h\setminus \partial \Omega \) as a common edge, denote \(v_0^1, v_0^2\) the values of \(v\) in the interior of \(T_1, T_2\), respectively. It follows from \(Q_b v_0^1=Q_b v_0^2=v_b\) on edge \(e\) and the fact \(\nabla v_0\cdot {\mathbf {n}}_e=v_n\in P_{k-1}(e)\) that
where \(\mathbf n _{T_1}, \mathbf n _{T_2}\) denote the outward unit normal vectors on \(e\) according to elements \(T_1, T_2\), respectively. This, together with \(\nabla v_0\cdot \mathbf n =v_n=0\) on the boundary edge \(e\in \mathcal {E}_h\cap \partial \Omega \) implies
It follows from Eq. (3.9) that \(\Vert \nabla v_0\Vert _T=0\) on each element \(T\). Thus, \(v_0= const\) locally on each element and is then continuous across each interior edge \(e\) as
The boundary condition of \(v_b=0\) then implies that \(v\equiv 0\) on \(\Omega \), which completes the proof. \(\square \)
Lemma 3.3
The weak Galerkin finite element scheme (3.5) has a unique solution.
Proof
Assume \(u_h^{(1)}\) and \(u_h^{(2)}\) are two solutions of the WG finite element scheme (3.5). It is obvious that the difference \(\rho _h = u_h^{(1)}- u_h^{(2)}\) is a finite element function in \(V_h^0\) satisfying
By letting \(v=\rho _h\) in above Eq. (3.10) we obtain the following indentity
It follows from Lemma 3.2 that \(\rho _h\equiv 0\), which shows that \(u_h^{(1)}=u_h^{(2)}\). This completes the proof. \(\square \)
4 An Error Estimate
The goal of this section is to establish an error estimate for the WG-FEM solution \(u_h\) arising from (3.5).
First of all, let us derive an error equation for the WG finite element solution obtained from (3.5). This error equation is critical in convergence analysis.
Lemma 4.1
Let \(u\) and \(u_h \in V_h\) be the solution of (1.1)–(1.3) and (3.5), respectively. Denote by
the error function between the \(L^2\) projection of \(u\) and its weak Galerkin finite element solution. Then the error function \(e_h\) satisfies the following equation
for all \(v\in V^0_h\). Here
Proof
Using (2.4) with \(\varphi =\Delta _{\omega }Q_hu = \mathbb {Q}_h\Delta u\) we obtain
which implies that
Next, it follows from the integration by parts that
By summing over all \(T\) and then using the identity \((\Delta ^2u,v_0)=(f, v_0)\) we arrive at
where we have used the fact that \(v_n\) and \(v_b\) vanish on the boundary of the domain. Combining the above equation with (4.3) yields
Adding \(s(Q_hu, v)\) to both sides of the above equation gives
Subtracting (3.5) from (4.4) leads to the following error equation
for all \(v\in V^0_h\). This completes the derivation of (4.1). \(\square \)
The following Theorem presents an optimal order error estimate for the error function \(e_h\) in the trip-bar norm. We believe this tripe-bar norm provides a discrete analogue of the usual \(H^2\)-norm.
Theorem 4.2
Let \(u_h\in V_h\) be the weak Galerkin finite element solution arising from (3.5) with finite element functions of order \(k\ge 2\). Assume that the exact solution of (1.1)–(1.3) is sufficiently regular such that \(u\in H^{k+2}(\Omega )\). Then, there exists a constant \(C\) such that
The above estimate is of optimal order in terms of the meshsize \(h\), but not in the regularity assumption on the exact solution of the biharmonic equation.
Proof
By letting \(v =e_h\) in the error Eq. (4.1), we have
where
The rest of the proof shall estimate each of the terms on the right-hand side of (4.7). For the first term, we use the Cauchy–Schwarz inequality and the estimates (7.5) and (7.6) in Lemma 7.4 (see “Appendix”) with \(m = k\) to obtain
For the second term, using Lemmas 7.4, 7.6 and 7.9 we obtain
where the \(H^{k+2}\)-norm of \(u\) is used because the estimate in Lemma 7.9 is not optimal in terms of the mesh parameter \(h\).
The third and fourth terms can be estimated by using the Cauchy–Schwarz inequality and the estimates (7.7) and (7.8) in Lemma 7.4 as follows
and
Substituting (4.8)–(4.11) into (4.6) gives
which implies (4.5) and hence completes the proof. \(\square \)
5 Error Estimates in \(L^2\)
In this section, we shall establish some error estimates for all three components of the error function \(e_h\) in the standard \(L^2\) norm.
First of all, let us derive an error estimate for the first component of the error function \(e_h\) by applying the usual duality argument in the finite element analysis. To this end, we consider the problem of seeking \(\varphi \) such that
Assume that the dual problem has the \(H^4\) regularity property in the sense that the solution function \(\varphi \in H^4\) and there exists a constant \(C\) such that
Theorem 5.1
Let \(u_h\in V_h\) be the weak Galerkin finite element solution arising from (3.5) with finite element functions of order \(k\ge 2\). Let \(k_0=\min \{3, k\}\). Assume that the exact solution of (1.1)–(1.3) is sufficiently regular such that \(u\in H^{k+2}(\Omega )\) and the dual problem (5.1) has the \(H^4\) regularity. Then, there exists a constant \(C\) such that
which means we have a sub-optimal order of convergence for \(k=2\) and optimal order of convergence for \(k\ge 3\).
Proof
Testing (5.1) by error function \(e_0\) and then using the integration by parts gives
where we have used the fact that \(e_n\) and \(e_b\) vanishes on the boundary of the domain \(\Omega \). By letting \(u=\varphi \) and \(v_0=e_h\) in (4.3), we can rewrite the above equation as follows
Next, by letting \(v=Q_h \varphi \), from the error equation (4.1), we have
Combining the two equations above gives
From the Cauchy–Schwarz inequality and Lemma 7.4, we can estimate the six terms on the right-hand side of the identity above as follows.
For the first term, it follows from Lemmas 7.4, 7.9 and the fact \(k_0=\min \{k, 3\}\le 3\) that
For the second term, it follows from (7.3) with \(m=k_0\) that
As to the third term, it follows from Cauchy–Schwarz inequality and Lemma 7.4 that
For the forth term, by using Lemma 7.3, we have
As to the fifth term, we also use the Cauchy–Schwarz inequality and Lemma 7.4 to obtain
The last term can be estimated as follows
Substituting all the six estimates into (5.4) we obtain
Using the regularity estimate (5.2) and choosing constant \(\lambda \) such that \(C \lambda \Vert \varphi \Vert _4<\frac{1}{2}\Vert e_0\Vert \), we arrive at
Together with the \(H^2\) error estimate (4.5) we have the desired \(L^2\) error estimate (5.3). \(\square \)
In order to study the error estimates on edges, we shall introduce the edge-based \(L^2\) norm here. To keep the consistency of order, the edge-based \(L^2\) norm is different from the standard \(L^2\) norm.
Definition 5.2
For any function \(v\) defined on the edges \(\mathcal {E}_h\),
where \(h_e\) is the measure of edge \(e\in \mathcal {E}_h\).
Next, we shall derive the estimates for the second and third components of the error function \(e_h\).
Theorem 5.3
Let \(u_h\in V_h\) be the weak Galerkin finite element solution arising from (3.5) with finite element functions of order \(k\ge 2\). Let \(k_0=\min \{k,3\}\). Assume that the exact solution of (1.1)–(1.3) is sufficiently regular such that \(u\in H^{k+2}(\omega )\) and the dual problem (5.1) has the \(H^4\) regularity property. Then, there exists a constant \(C\) such that
Proof
It is obvious that
Summing over all edges, we have
We shall discuss the two terms separately. For the first part, by applying the trace inequality (7.1), the inverse inequality (7.2) and the error estimate for \(e_0\) in Theorem 5.1, we have
For the second part, we use the trip-bar norm to handle the second part.
Combining the above two estimates gives the desired error estimate (5.11).
Similarly, we establish the error estimates for \(e_n\).
Thus, we have
which completes the proof. \(\square \)
6 Numerical Results
In this section, we would like to report some numerical results for the weak Galerkin finite element method proposed and analyzed in previous sections. Here we use the following finite element space
For any given \(v=\{v_0, v_b, v_n \mathbf n _e \}\in \tilde{V}_h\) and \(\varphi \in P_0(T)\), we compute the discrete weak Laplacian \(\Delta _w v\) on each element \(T\) as a function in \(P_0(T)\) as follows
which could be simplified as
The error for the weak Galerkin solution is measured in six norms defined as follows:
Example 6.1
Consider the biharmonic problem (1.1)–(1.3) in the square domain \(\Omega =(0,1)^2\). It has the analytic solution \(u(x)=x^2(1-x)^2y^2(1-y)^2\), and the right hand side function \(f\) in (1.1) is computed to match the exact solution. The mesh size is denoted by \(h=1/n\). Table 1 shows that the convergence rates for the WG-FEM solution in the \(H^2\) and \(L^2\) norms are of order \(O(h)\) and \(O(h^2)\) when \(k=2\), respectively.
Table 2 shows that the errors and orders of Example 6.1 in \(L^2\) and \(L^\infty \) for \(e_b\). The numerical results are in consistency with theory for these two cases.
Table 3 shows that the errors and orders of Example 6.1 in \(L^2\) and \(L^\infty \) for \(e_n\). The numerical results are in consistency with theory for these two cases.
In Tables 4, 5 and 6 we investigate the same problem for \(k = 3\). Table 4 shows that the convergence rates for the WG-FEM solution in the \(H^2\) and \(L^2\) norms are of order \(O(h^2)\) and \(O(h^4)\). Tables 5 and 6 show the errors and orders in \(L^2\) and \(L^\infty \) for \(e_b\) and \(e_n\), which are also consistent with theoretical conclusions.
Example 6.2
Consider the biharmonic problem (1.1)–(1.3) in the square domain \(\Omega =(0,1)^2\). It has the analytic solution \(u(x)=\sin (\pi x)\sin (\pi y)\), and the right hand side function \(f\) in (1.1) is computed accordingly.
The numerical results are presented in Tables 7, 8, 9, 10, 11 and 12 which confirm the theory developed in previous sections.
References
Arad, M., Yakhot, A., Ben-Dor, G.: A highly accurate numerical solution of a biharmonic equation. Numer. Methods Partial Differ. Equ. 13, 375C391 (1998)
Argyris, J.H., Fried, I., Scharpf, D.W.: The TUBA family of plate elements for the matrix displacement method. Aeronaut. J. R. Aeronaut. Soc. 72, 514C517 (1968)
Argyris, J.H., Dunne, P.C.: The finite element method applied to fluid dynamics. In: Hewitt, B.L., Illingworth, C.R., Lock, R.C., Mangler, K.W., McDonnel, J.H., Richards, C., Walkden, F. (eds.) Computational Methods and Problems in Aeronautical Fluid Dynamics, pp. 158–197. Academic Press, London (1976)
Arnold, D.N., Brezzi, F.: Mixed and nonconforming finite element methods: implementation, postprocessing and error estimates. RAIRO Modl. Math. Anal. Numr. 19(1), 7–32 (1985)
Behrens, E.M., Guzmán, J.: A mixed method for the biharmonic problem based on a system of first-order equations. SIAM J. Numer. Anal. 49, 789–817 (2011)
Bialecki, B., Karageorghis, A.: A Legendre spectral Galerkin method for the biharmonic Dirichlet problem. SIAM J. Sci. Comput. 22(5), 1549–1569 (2000)
Bialecki, B., Karageorghis, A.: Spectral Chebyshev collocation for the Poisson and biharmonic equations. SIAM J. Sci. Comput. 32(5), 2995–3019 (2010)
Bialecki, B.: A fast solver for the orthogonal spline collocation solution of the biharmonic Dirichlet problem on rectangles. J. Comput. Phys. 191, 601–621 (2003)
Bjorstad, P.: Fast numerical solution of the biharmonic dirichlet problem on rectangles. SIAM J. Numer. Anal. 20, 59–71 (1983)
Brenner, S., Sung, L.: \(C^0\) interior penalty methods for fourth order elliptic boundary value problems on polygonal domains. J. Sci. Comput. 22/23, 83–118 (2005)
Chan, R.H., DeLillo, T.K., Horn, M.A.: The numerical solution of the biharmonic equation by conformal mapping. SIAM J. Sci. Comput. 18, 1571–1582 (1997)
Chen, G., Li, Z., Lin, P.: A fast finite difference method for biharmonic equations on irregular domains and its application to an incompressible Stokes flow. Adv. Comput. Math. 29, 113–133 (2008)
Chen, H.R., Chen, S.C., Qiao, Z.H.: \(C^0\)-nonconforming tetrahedral and cuboid elements for the three-dimensional fourth order elliptic problem. Numer. Math. 124(1), 99–119 (2013)
Chen, H.R., Chen, S.C., Qiao, Z.H.: \(C^0\)-nonconforming triangular prism elements for the three-dimensional fourth order elliptic problem. J. Sci. Comput. 55(3), 645–658 (2013)
Ciarlet, P.A., Raviart, P.G.: A mixed finite element method for the biharmonic equation. In: Mathematical Aspects of Finite Elements in Partial Differential Equations. Academic Press, New York, pp. 125–145 (1974)
Clough, R.W., Tocher, J.L.: Finite element stiffness matrices for analysis of plates in bending. In: Proceedings of the Conference on Matrix Methods in Structural Mechanics. Wright Patterson A.F.B., Ohio (1965)
Cockburn, B., Dong, B., Guzmán, J.: A hybridizable and superconvergent discontinuous Galerkin method for biharmonic problems. J. Sci. Comput. 40, 141C187 (2009)
Davini, C., Pitacco, I.: An unconstrained mixed method for the biharmonic problem. SIAM J. Numer. Anal. 38, 820C836 (2001)
Dean, E.J., Glowinski, R., Pironneau, O.: Iterative solution of the stream functionvorticity formulation of the Stokes problem, applications to the numerical simulation of incompressible viscous flow. Comput. Methods Appl. Mech. Eng. 87, 117–155 (1991)
Ehrlich, L.W., Gupta, M.M.: Some difference schemes for the biharmonic equation. SIAM J. Numer. Anal. 12, 773–790 (1975)
Engel, G., Garikipati, K., Hughes, T., Larson, M.G., Mazzei, L., Taylor, R.: Continuous/discontinuous finite element approximations of fourth order elliptic problems in structural and continuum mechanics with applications to thin beams and plates, and strain gradient elasticity. Comput. Meth. Appl. Mech. Eng. 191, 3669–3750 (2002)
Fraeijs de Veubeke, B.: A conforming finite element for plate bending. In: Zienkiewicz, O.C., Holister, G.S. (eds.) Stress Analysis, p. 145C197. Wiley, New York (1965)
Greenbaum, A., Greengard, L., Mayo, A.: On the numerical solution of the biharmonic equation in the plane. Phys. D 60, 216–225 (1992)
Gupta, M.M., Manohar, R.P.: Direct solution of biharmonic equation using noncoupled approach. J. Comput. Phys. 33, 236–248 (1979)
Heinrichs, W.: A stabilized treatment of the biharmonic operator with spectral methods. SIAM J. Sci. Stat. Comput. 12, 1162–1172 (1991)
Jaswon, M.A., Symm, G.T.: Integral Equation Methods in Potential Theory and Elastostatics, pp. 99–126. Academic Press, New York (1977)
Linden, J.: A Multigrid Method for Solving the Biharmonic Equation on Rectangular Domains, Notes Numer. Fluid Mech. 11, Vieweg, Braunschweig (1985)
Morley, L.S.D.: The triangular equilibrium element in the solution of plate bending problems. Aero. Quart. 19, 149–169 (1968)
Mayo, A.: The fast solution of Poissons and the biharmonic equations on irregular regions. SIAM J. Numer. Anal. 21, 285–299 (1984)
Mozolevski, I., Sli, E., Bsing, P.R.: hp-Version a priori error analysis of interior penalty discontinuous Galerkin finite element approximations to the biharmonic equation. J. Sci. Comput. 30, 465–491 (2007)
Mu, L., Wang, J., Ye, X.: Weak Galerkin finite element methods on polytopal meshes. arXiv:1204.3655v2
Mu, L., Wang, J., Ye, X.: Weak Galerkin finite element methods for the biharmonic equation on polytopal meshes, preprint
Mu, L., Wang, J., Ye, X., Zhang, S.: A \(C^0\)-weak Galerkin finite element method for the biharmonic euqation, preprint
Muskhelishvili, N.I.: Some Basic Problems in the Mathematical Theory of Elasticity. Noordhoff Groningen, The Netherlands (1953)
Pandit, S.K.: On the use of compact streamfunction-velocity formulation of steady Navier–Stokes equations on geometries beyond rectangular. J. Sci. Comput. 36, 219–242 (2008)
Roache, P.: Computational Fluid Dynamics. Hermosa, Albuquerque (1972)
Wang, C., Wang, J.: An efficient numerical scheme for the biharmonic equation by weak Galerkin finite element methods on polygonal or polyhedral meshes. arXiv:1303.0927
Wang, J., Ye, X.: A weak Galerkin finite element method for second-order elliptic problems. J. Comput. Appl. Math. 241, 103–115 (2013)
Wang, J., Ye, X.: A weak Galerkin mixed finite element method for second-order elliptic problems. Math. Comput. 83(289), 2101–2126 (2014)
Wang, J., Ye, X.: A weak Galerkin finite element method for the Stokes equations. arXiv:1302.2707v1
Zlámal, M.: On the finite elmeent method. Numer. Math. 12, 394–409 (1968)
Acknowledgments
We gratefully acknowledge Professor Junping Wang for presenting this problem and giving us many valuable suggestions. The authors also thank the anonymous referees and editor for their careful reading of the manuscript and their valuable comments to improvement the work.
Author information
Authors and Affiliations
Corresponding author
Additional information
The research of Zhang was supported in part by China Natural National Science Foundation (11271157, 11371171, 11471141), and by the Program for New Century Excellent Talents in University of Ministry of Education of China.
Appendix: \(L^2\) Projection and Some Technical Results
Appendix: \(L^2\) Projection and Some Technical Results
In this section, we shall present some technical results for the \(L^2\) projection operators with respect to the finite element space \(V_h\). These results are useful for the error estimates of the WG finite element method.
Lemma 7.1
([39]) (Trace Inequality) Let \(\mathcal {T}_h\) be a partition of the domain \(\Omega \) into polygons in 2D or polyhedra in 3D. Assume that the partition \(\mathcal {T}_h\) satisfies the assumptions (A1), (A2), and (A3) as specified in [39]. Then, there exists a constant \(C\) such that for any \(T\in \mathcal {T}_h\) and edge/face \(e\in \partial T\), we have
where \(\theta \in H^{1}(T)\) is any function.
Lemma 7.2
([39]) (Inverse Inequality) Let \(\mathcal {T}_h\) be a partition of the domain \(\Omega \) into polygons or polyhedra. Assume that \(\mathcal {T}_h\) satisfies all the assumptions (A1)–(A4) as specified in [39]. Then, there exists a constant \(C(n)\) such that
for any piecewise polynomial \(\varphi \) of degree \(n\) on \(\mathcal {T}_h\).
1.1 Approximation Properties
The following lemma provides some approximation properties for the projection operators \(Q_h\) and \(\mathbb {Q}_h\).
Lemma 7.3
([32]) Let \(\mathcal {T}_h\) be a finite element partition of \(\Omega \) satisfying the shape regularity assumptions. Then, for any \(0\le s\le 2\) and \(2\le m\le k\) we have
Lemma 7.4
Let \( 2 \le m\le k, \omega \in H^{m+2}(\Omega )\). There exists a constant \(C\) such that the following estimates hold true:
Here \(\delta _{i,j}\) is the usual Kronecker’s delta with value \(1\) when \(i=j\) and value 0 otherwise.
Proof
To derive (7.5), we use the trace inequality (7.1) and the estimate (7.4) to obtain
As to (7.6), we use the trace inequality (7.1) and the estimate (7.4) to obtain
As to (7.7), we have from the definition of \(Q_b \), the trace inequality (7.1), and the estimate (7.3) that
Notice that \(Q_b\) is a linear bounded operator, we use the definition of \(Q_b\) and the trace inequality (7.1) to obtain
To derive (7.9), we use the trace inequality (7.1) and the estimate (7.4) to obtain
This completes the proof of (7.9), and hence the lemma. \(\square \)
1.2 Technical Inequalities
The goal here is to present some technical estimates useful for deriving error estimates for the WG finite element scheme (3.5).
Lemma 7.5
There exists a constant \(C\) such that, for any \(v=\{v_0, v_b,v_n\mathbf n _e\}\in V_h\), the following holds true
Proof
From the identity (2.4) with \(\phi =\Delta v_0\) we have
Thus, using the Cauchy–Schwarz inequality, trace inequality, and the inverse inequality we obtain
Hence,
which verifies the inequality (7.10). \(\square \)
Lemma 7.6
([37], Lemma 10.4) There exists a constant \(C\) such that, for any \(v\in V_h^0\), we have the following Poincaré inequality:
The following lemma provides an estimate for the term \(\sum _{T\in {\mathcal {T}}_h}\Vert \nabla v_0\Vert _T^2\). Note that \(v_0\) is a piecewise polynomial of degree \(k\ge 2\). Thus, Lemma 7.7 is concerned only with piecewise polynomials; no boundary condition is necessary.
Lemma 7.7
Let \(\varphi \) be any piecewise polynomial of degree \(k\ge 2\) on each element \(T\). Denote by \(\nabla _h\varphi \) and \(\Delta _h \varphi \) the gradient and Laplacian of \(\varphi \) taken on each element. Then, for any \(\varepsilon >0\), there exists a constant \(C\) such that
Here \(\varphi _L\) is the trace of \(\varphi \) on \(e\) as seen from the “left” or the opposite direction of \({\mathbf {n}}_e\). If \(e\) is a boundary edge, then the trace from the outside of \(\Omega \) is defined as zero.
Proof
On each element \(T\), we have
Summing over all \(T\in \mathcal {T}_h\), we have
Using the identity \(a_L b_L+a_R b_R=(a_L+a_R)b_{L}+a_R(b_R-b_L)\) we obtain
Thus, from the Cauchy–Schwarz inequality we have
Next, we use the trace inequality (7.1) and the inverse inequality (7.2) to obtain
and
Substituting (7.15) and (7.16) into (7.14) yields
Substituting (7.17) into (7.13) gives
which, through an use of Young’s inequality, implies the desired estimate (7.12). This completes the proof. \(\square \)
Lemma 7.8
There exists a constant \(C\) such that for any \(v=\{v_0,v_b,v_n{\mathbf {n}}_e\}\in V_h^0\) the following Poincaré type inequality holds true
In addition, we have the following estimate
where \(\lambda \) is a positive constant.
Proof
The first component \(v_0\) is a piecewise polynomial of degree \(k\ge 2\). Using the estimate (7.12) in Lemma 7.7 we have
By inserting \(v_n{\mathbf {n}}_e\cdot {\mathbf {n}}\) in each integrand we obtain
Similarly, by inserting \(v_b\)
Substituting the above two inequalities into (7.20) yields
Using the Poincaré inequality (7.11) and the estimate (7.10) we arrive at
which leads to the inequality (7.18) for sufficiently small \(\varepsilon \).
Finally, by setting \(\varepsilon = \lambda h^{-2}\) in (7.21) we arrive at
where \(\lambda \) is a positive constant. This verifies the inequality (7.19), and hence completes the proof of the lemma. \(\square \)
Lemma 7.9
There exists a constant \(C\) such that for any \(v=\{v_0,v_b,v_n{\mathbf {n}}_e\}\in V_h^0\) one has
and
Proof
From the trace inequality (7.1) and the inverse inequality (7.2), we have
Summing over all \(T\in \mathcal {T}_h\) yields
which, combined with (7.18) and (7.19), completes the proof of the lemma. \(\square \)
Remark 7.1
The estimate (7.22) in Lemma 7.9 is sufficient for us to derive an optimal order error estimate for the WG finite element solution arising from (3.5). But the estimate (7.22) is sub-optimal in terms of the mesh parameter \(h\). We conjecture that the following inequality holds true
However, with the current mathematical approach, we are unable to verify the validity of (7.25). This estimate is then left to interested readers or researchers as an open problem.
Rights and permissions
About this article
Cite this article
Zhang, R., Zhai, Q. A Weak Galerkin Finite Element Scheme for the Biharmonic Equations by Using Polynomials of Reduced Order. J Sci Comput 64, 559–585 (2015). https://doi.org/10.1007/s10915-014-9945-7
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10915-014-9945-7