Abstract
We propose a mixed numerical method for solving the compressible Navier–Stokes system and study its convergence and stability with respect to the physical domain. The numerical solutions are shown to converge, up to a subsequence, to a weak solution of the problem posed on the limit domain.
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1 Introduction
There is a great theoretical and evidently also practical interest in the problem of convergence of numerical methods used for simulation of fluids in continuum mechanics. Ignoring the influence of temperature changes we consider a mathematical model of a compressible, barotropic, viscous fluid occupying a bounded physical domain \(\Omega \subset R^3\). In the Eulerian coordinate system, the time evolution of the fluid is described by means of the mass density \(\varrho = \varrho (t,x)\) and the velocity field \(\mathbf{u}= \mathbf{u}(t,x)\), \(t \in (0,T)\), \(x \in \Omega \), governed by the Navier–Stokes system of equations:
where \(p = p(\varrho )\) is the pressure, and the symbol \(\mathbb {S}(\nabla _x\mathbf{u})\) denotes the viscous stress tensor, here determined by Newton’s rheological law:
The barotropic pressure \(p = p(\varrho )\) is a continuously differentiable function of the density satisfying
Remark 1.1
The condition \(\gamma > 3\) is technical; the so-called adiabatic exponent for real fluids ranges in the interval \(\gamma \in (1, 5/3]\), where the extremal value \(\gamma = 1\) corresponds to the isothermal case, while \(\gamma = 5/3\) characterizes the monoatomic gas.
Remark 1.2
Since the viscosity coefficients \(\mu \) and \(\eta \) are constant, we may write
The system is supplemented with the standard no-slip boundary condition
and the initial conditions
Remark 1.3
We deliberately omitted the action of an external force to simplify the presentation. As will become clear in what follows, a bounded driving force can be incorporated in the system with only minor modifications of the proof of convergence.
1.1 Weak solutions
We adopt the standard weak formulation of the problem (1.1)–(1.7).
Definition 1.1
We say that \([\varrho , \mathbf{u}]\) is a weak solution to the problem (1.1)–(1.7) in \((0,T) \times \Omega \) if:
-
$$\begin{aligned}&\displaystyle \varrho \ge 0\quad \text{ a.a. } \text{ in }\ (0,T) \times \Omega , \ \varrho \in L^\infty (0,T; L^\gamma (\Omega )), \ \mathbf{u}\in L^2(0,T; W^{1,2}_0(\Omega ;R^3)),\nonumber \\ \end{aligned}$$(1.8)$$\begin{aligned}&\displaystyle p(\varrho ) \in L^1((0,T) \times \Omega ),\ \varrho \mathbf{u}\in L^\infty (0,T; L^{\frac{2 \gamma }{\gamma + 1}}(\Omega ;R^3)); \end{aligned}$$(1.9)
-
$$\begin{aligned} \int _0^T \int _{\Omega } \Big [ \varrho \partial _t \varphi + \varrho \mathbf{u}\cdot \nabla _x\varphi \Big ] \ \mathrm{d} {x} \ \mathrm{d} t = - \int _{\Omega } \varrho _0 \varphi (0, \cdot ) \ \mathrm{d} {x} \end{aligned}$$(1.10)
for any \(\varphi \in C^\infty _c([0,T) \times \overline{\Omega })\);
-
$$\begin{aligned}&\int _0^T \int _{\Omega } \Big [ \varrho \mathbf{u}\cdot \partial _t \varphi + \varrho \mathbf{u}\otimes \mathbf{u}: \nabla _x\varphi + p(\varrho ) \mathrm{div}_x\varphi \Big ] \ \mathrm{d} {x} \ \mathrm{d} t \nonumber \\&\quad = \int _0^T \int _{\Omega } \Big [ \mu \nabla _x\mathbf{u}: \nabla _x\varphi + \lambda \mathrm{div}_x\mathbf{u}\ \mathrm{div}_x\varphi \Big ] \ \mathrm{d} {x} \ \mathrm{d} t - \int _{\Omega } \varrho _0 \mathbf{u}_0 \cdot \varphi (0, \cdot ) \ \mathrm{d} {x}\nonumber \\ \end{aligned}$$(1.11)
for any \(\varphi \in C^\infty _c([0,T) \times {\Omega }; R^3)\);
-
the energy inequality
$$\begin{aligned}&\int _{\Omega } \left[ \frac{1}{2} \varrho |\mathbf{u}|^2 + P(\varrho ) \right] (\tau , \cdot ) \ \mathrm{d} {x} + \int _0^\tau \int _{\Omega } \left[ \mu |\nabla _x\mathbf{u}|^2 + \lambda |\mathrm{div}_x\mathbf{u}|^2 \right] \ \mathrm{d} {x} \ \mathrm{d} t \nonumber \\&\quad \le \int _{\Omega } \left[ \frac{1}{2} \varrho _0 |\mathbf{u}_0|^2 + P(\varrho _0) \right] \ \mathrm{d} {x}, \quad \text{ with }\ P(\varrho ) = \varrho \int _1^\varrho \frac{p(z)}{z^2} \ \mathrm{d}z, \end{aligned}$$(1.12)holds for a.a. \(\tau \in (0,T)\).
The existence of global-in-time weak solutions under the hypothesis \(\gamma \ge 9/5\) in (1.4) was proved by Lions [20]. The result was later extended to the range \(\gamma > 3/2\) in [13]. Unfortunately, the proof of existence in the “subcritical” range \(\gamma \le 3\) consists of at least two steps performed at different level of approximations and as such therefore not directly transferable to the numerical setting.
1.2 Numerical method
Our goal is to propose a numerical method for solving the Navier–Stokes system (1.1–1.7) and to show its stability with respect to the underlying spatial domain and convergence towards a weak solution specified in Definition 1.1. To this end, we adapt the discontinuous Galerkin finite element scheme proposed in [17, 18] for the compressible Navier–Stokes system.
Since we are interested in smooth spatial domains, we consider an unfitted mesh on a family of polyhedral domains \(\{ \Omega _h \}_{h > 0}\) approximating the target physical space \(\Omega \) in the following sense: For any compact \(K_i \subset \Omega \) it holds
and, similarly, for any compact \(K_e \subset R^3 {\setminus } \overline{\Omega }\),
Besides the relatively straightforward modifications to accommodate the case of variable numerical domain, we also introduce a new “dissipative” discretization implemented in the upwind terms. In such a way, we eliminate completely the artificial viscosity regularization used by several authors (see e.g. Eymard et al. [9]) including the original scheme proposed in [18]. Very roughly indeed, this new approach may be compared to adding an artificial viscosity to both equations in (1.1, 1.2):
where the artificial viscosity is active only for small values of the velocity amplitude \(|\mathbf{u}|\). The resulting “dissipative” upwind operator remains therefore much closer to the approximated convective terms in the continuous equations.
We note that the fact that the limit problem is defined on a possibly smooth domain may be of interest when establishing convergence of the scheme. The problem (1.1)–(1.7) is known to possess a local regular solution that can be extended to the full time interval (0, T) as soon as we control the amplitude of the density, see Sun, Wang, and Zhang [21]. Moreover, any weak solution coincides with the strong solution as long as the latter exists, see [11]. Consequently, boundedness of the numerical densities implies unconditional convergence as long as the domain \(\Omega \) is sufficiently smooth, see Sect. 8 for details.
The paper is organized as follows. In Sect. 2, we introduce the necessary numerical framework including the basic notation and several useful properties of the underlying function spaces. The numerical scheme is introduced in Sect. 3, where we also state our main result concerning convergence towards a weak solution of the Navier–Stokes system. In Sect. 4, we derive a renormalized version of the continuity equation as well as the discrete version of the total energy balance. Section 5 is devoted to the stability of the scheme, containing the uniform bounds necessary for the limit passage. In Sect. 6, we discuss the problem of consistency of the method rewriting finally the numerical scheme in terms of the standard weak formulation based on smooth test functions. Having established consistency, we show convergence of the scheme by adapting the steps of [10, Chapter 7]. Here, similarly to the existence theory, the key idea is the weak continuity property of the effective viscous flux discovered by Lions [20]. Finally, we discuss the implications of some recent results concerning the weak-strong uniqueness property and regularity of the weak solutions on the problem of unconditional convergence of the numerical scheme in Sect. 8.
2 Preliminaries
In this section, we collect the necessary material from numerical analysis. For two numerical quantities a, b, we shall write
Here, “constant” typically means a generic quantity independent of the size of the mesh and the time step used in the numerical scheme as well as other parameters as the case may be.
2.1 Mesh
We suppose that the numerical domains \(\Omega _h\) admit a tetrahedral mesh \(E_h\); the individual elements in the mesh will be denoted by \(E \in E_h\). Faces in the mesh are denoted by \(\Gamma \), whereas \(\Gamma _h\) is the set of all faces. Moreover, the set of faces \(\Gamma \subset \partial \Omega _h\) is denoted by \(\Gamma _{h, \mathrm{ext}}\), while \(\Gamma _{h,\mathrm{int}} = \Gamma _h {\setminus } \Gamma _{h, \mathrm{ext}}\). The size (diameter of elements in the mesh) is proportional to a positive parameter h. For \(E, F \in E_h\), \(E \ne F\), the intersection \(E \cap F\) is either a vertex, or an edge, or a face \(\Gamma \in \Gamma _h\). The mesh is assumed to be shape regular, meaning that the radius of the circumsphere and the biggest ball inside each element are “\(\approx \)” proportional to h. Finally, the family \(\{ \Omega _h \}_{h > 0}\) will approximate a limit domain \(\Omega \subset R^3\) in the sense specified in (1.13, 1.14).
Each face \(\Gamma \in \Gamma _h\) is associated with a fixed normal vector \(\mathbf{n}\). On the other hand, we write \(\Gamma _E\) whenever a face \(\Gamma _E \subset \partial E\) is considered as a part of the boundary of the element E. In such a case, the normal vector to \(\Gamma _E\) is always the outer normal vector with respect to E. Keeping this convention in mind we introduce for any function g, continuous on each element E,
For \(\Gamma _E \subset \partial E\) we simply write g for \(g^\mathrm{in}\). Occasionally, we also omit the subscript \(\Gamma \) if no confusion arises.
2.2 Piecewise constant finite elements
We introduce the space
of piecewise constant functions along with the associated projection
we will occasionally denote
Finally, we recall various forms of (scaled) Poincaré’s inequality:
for any \(1 \le q < \infty \), in particular,
2.3 Crouzeix–Raviart finite elements
A differential operator D acting on the x-variable will be discretized as
The Crouzeix–Raviart finite element spaces (see Brezzi and Fortin [4], among others) are defined as
together with
Next, we introduce the associated projection
requiring
It is easy to check that
and
see [19, Lemma 2.11].
We also recall the the error estimates
for any \(v \in W^{m,q} (\Omega _h)\), see Crouzeix and Raviart [6], and [19, Lemma 2.7].
2.4 “Dissipative” upwind operator
Denoting
we introduce a dissipative upwind operator \(\mathrm{Up}[r, \mathbf{u}]\) on a face \(\Gamma \) in the form
with a positive exponent \(\alpha \) determined below. Note that such a definition makes sense as soon as \(r \in Q_h (\Omega _h)\), \(\mathbf{u}\in V_h(\Omega _h; R^3)\) and \(\Gamma \in \Gamma _{h, \mathrm{int}}\).
Setting, formally, \(h^\alpha \approx 0\) in (2.10), we obtain the conventional definition of the upwind operator
To illuminate the dissipative character of the new upwind operator, we may also write
where
Remark 2.1
The numerical diffusion supplied by the dissipative component is quite subtle; it acts only when \(| {\tilde{\mathbf{u}}}\cdot \mathbf{n}| < h^\alpha \) and has amplitude \(h^\alpha \). Note that the conventional artificial diffusion used by Eymard et al. [9] and [19] corresponds to
For \(r, F \in Q_h(\Omega _h)\), \(\mathbf{u}\in V_h(\Omega _h, R^3)\), \(\phi \in C^1(\overline{\Omega }_h)\), we may use Green’s theorem to compute
Furthermore, going back to (2.11) we deduce that
for any \(r,g \in Q_h(\Omega _h)\), \(\mathbf{u}\in V_{h,0}(\Omega _h; R^3)\). Finally, using formula (2.13), we may compute the first integral on the right-hand side of (2.12) for \(\mathbf{u}\in V_{h,0}(\Omega _h; R^3)\), specifically,
Thus, plugging the resulting expression in (2.12) we obtain a universal formula
for any \(r, F \in Q_h(\Omega _h)\), \(\mathbf{u}\in V_{h,0}(\Omega _h; R^3)\), \(\phi \in C^1(\overline{\Omega }_h)\).
2.5 \(L^p-L^q\) and trace estimates for finite elements
The estimates listed below are direct consequences of the assumed shape regularity of the mesh and follow by a scaling argument. We claim that
whence
where \(P_m\) denotes the space of polynomials of degree m.
Similarly,
and, making use of the inequality
with the summation over a finite index set for i, we finally obtain
We will also need a variant of (2.17) and (2.18) for the functions of the time variable \(t \in (0,T)\), where the discretization is of order \(\Delta t\). Evidently,
and, therefore
for any w that is constant on any time segment \([j\Delta t, (j+1) \Delta t]\) contained in [0, T].
Finally, we recall the estimate
that follows directly from Poincarè’s inequality (2.3).
2.6 Discrete Sobolev spaces
We introduce the discrete \(H^1\)-(semi)norm
for \(v \in Q_h(\Omega _h)\). We report the following estimate that may be seen as a discrete analogue of the well-known estimates for Sobolev functions in \(W^{1,2}\):
for any compact \(K_i \subset \Omega \), \(|\xi | < \mathrm{dist}[K_i, \partial \Omega _h]\), \(v \in Q_h(\Omega _h)\), see Eymard et al. [8, Section 5].
Remark 2.2
In view of our hypothesis (1.13), the expression on the left is defined provided \(h = h(K_i)\) is small enough.
Similarly, we may define a discrete \(H^1\)-norm on the space \(V_{h,0}(\Omega _h)\) by setting
In view of the limit passage \(\Omega _h \rightarrow \Omega \), it is convenient to extend a function \(v \in V_{h,0}(\Omega )\) to be zero outside \(\Omega \). With this convention, we have
and
for any \({v} \in V_{h,0}(\Omega _h)\), see Gallouët et al. [16].
Finally, the following assertion follows from (2.22), (2.25) and can be seen as a special case of the results of Christiansen, Munthe-Kaas and Owren [5, Proposition 5.67]:
Lemma 2.1
For any function \(v \in V_{h,0}(\Omega _h)\) there exists an \(R^V_h [v] \in C^\infty _c(R^3)\) such that
Moreover,
Similarly, for any \(g \in Q_h(\Omega _h)\) there is an \(R^Q_h [g] \in C^\infty (R^3)\) such that
for any compact \(K_i \subset \Omega \).
Remark 2.3
The regularizing operators \(R^V_h[v]\), \(R^Q_h[v]\) can be taken as a spatial convolution with a regularizing kernel, see [5] for details.
3 Numerical scheme, main result
Having collected the necessary material, we introduce the numerical scheme to solve the Navier–Stokes system (1.1)–(1.7).
3.1 Numerical scheme
We start by approximating the initial data by their projections onto the space \(Q_h(\Omega _h)\). To this end, we assume that both \(\varrho _0\) and \(\mathbf{u}_0\) are functions defined on the whole space \(R^3\) and set
Next, we introduce the discrete time derivative
and define successively the sequence of numerical solutions \([\varrho ^k_h, \mathbf{u}^k_h]_{h > 0}\), \(k=1,2,\ldots \),
satisfying:
Continuity method
Momentum method
Remark 3.1
We recall that \(\widehat{\mathbf{u}}^k_h = \Pi ^Q_h [\mathbf{u}^k_h]\) denotes the projection onto the space \(Q_h\) of piecewise constant functions. As we will see, our discretization of the convective term in (3.3), taken over from [19], yields a numerical analogue of the energy inequality providing the necessary stability estimates.
3.2 Main result
Before stating our main result, it is convenient to extend the numerical solution to be defined for any \(t \ge 0\). To this end, we set
Accordingly, we set
Besides, we also frequently use the already introduced convention that the functions in \(V_{h,0}(\Omega _h)\) are defined on the whole space \(R^3\), being extended to be zero outside \(\Omega _h\).
Our main result may be stated as follows:
Theorem 3.1
Let \(\Omega \subset R^3\) be a bounded Lipschitz domain approximated by a family of polyhedral domains \(\{ \Omega _h \}_{h > 0}\) as in (1.13, 1.14), where each \(\Omega _h\) admits a tetrahedral mesh satisfying the hypotheses specified in Sect. 2.1. Let \(\mu > 0\), \(\lambda \ge 0\), and let the pressure \(p = p(\varrho )\) satisfy the hypothesis (1.4) with
Let \([ \varrho _h, \mathbf{u}_h ]_{h > 0}\) be a family of numerical solutions constructed by means of the method (3.1)–(3.3) such that
with
where \(\alpha \) is the exponent in the dissipative upwinding (2.11).
Then, extending \(\varrho _h\), \(\mathbf{u}_h\) to be zero outside \(\Omega _h\) we have, at least for a suitable subsequence,
where \([\varrho , \mathbf{u}]\) is a weak solution of the problem (1.1)–(1.7) in \((0,T) \times \Omega \) in the sense of Definition 1.1.
Remark 3.2
As a matter of fact, the assumption that \(\Omega \) is Lipschitz is not really necessary and can be considerably relaxed, see [14]. It is enough to assume that the limit domain enjoys the so-called segment property, meaning that each point on the boundary \(\partial \Omega \) is an endpoint of a segment of fixed length, the interior of which is contained in \(R^3 {\setminus } \overline{\Omega }\).
Remark 3.3
The existence of the numerical solutions \([\varrho _h, \mathbf{u}_h]\) can be shown by means of a fixed point argument exactly as in [19].
Remark 3.4
The assumption \(p'(0) > 0\) facilitates the analysis but can also be relaxed, see [19].
4 Renormalization and the total energy balance
We introduce a renormalized variant of the continuity method (3.2) and derive a discrete analogue of the total energy balance (1.12). In what follows we use the notation
4.1 Renormalized equation of continuity
Take \(\phi = b'(\varrho ^k_h)\), where b is a smooth function, as a test function in the continuity method (3.2) to obtain
for a certain \(\xi ^{k}_h \in \mathrm{co}\{ \varrho ^{k-1}_h, \varrho ^{k}_h \}\).
Similarly to (2.13), the upwind term can be written as
Thus, summing up the previous estimates we obtain the integrated renormalized continuity method:
with
4.2 Energy inequality
Our goal is to derive a discrete counterpart of the energy inequality (1.12). To this end, we take \(\phi = \mathbf{u}^k_h\) as a test function in the momentum method (3.3). First, in accordance with the renormalized continuity method (4.1), we claim that
where the pressure potential P has been introduced in (1.12).
Next, we compute
The upwind term reads
Summing up (4.2)–(4.4) and making use of the continuity method (3.2) we deduce the energy inequality
with \(A = \inf _{\varrho > 0} \left\{ P''(\varrho ) \right\} \). Since \(P''(\rho ) = p'(\rho )\slash \rho \), we have \(A>0\) according to (1.4).
5 Stability
In this section, we derive uniform bounds for the family \([ \varrho _h, \mathbf{u}_h ]_{h > 0}\) independent of the time step \(\Delta t \approx h\) and the element size h.
5.1 Mass conservation
Taking \(\phi \equiv 1\) in the continuity method (3.2) we obtain
meaning that the total mass is conserved by the scheme.
5.2 Energy bounds
The energy inequality (4.5) yields
whence
and
whence, in accordance with (2.24),
where the bounds are uniform for \(h \rightarrow 0\). We recall that \(\mathbf{u}_h\) as well as other quantities, extended to be zero outside the numerical domain \(\Omega _h\), may be regarded as functions on the whole space \(R^3\).
Finally, we record the bounds resulting from numerical dissipation :
and
6 Consistency formulation
Having collected all the available uniform bounds, our next task is to verify that our numerical method is consistent with the variational formulation of the original problem.
6.1 Continuity method
For \(\phi \in C^\infty _c(R^3)\), take \(\Pi ^Q_h[\phi ]\) as a test function in the continuity method (3.1). Using the formula (2.14) for \(r = \varrho ^k_h\), \(\mathbf{u}= \mathbf{u}^k_h\), \(F = \Pi ^Q_h[\phi ]\) we check without difficulty that
Note that here
as \(\mathrm{div}_h\mathbf{u}^k_h\) is constant on each element E.
Now, by Hölder’s inequality,
where the first integral on the right-hand side is controlled in \(L^2(0,T)\) by (5.10).
As for the second integral, we may apply Hölder’s inequality, combined with Poincaré’s inequality (2.4) and the trace estimates (2.15), (2.16) to obtain
Finally, we use the interpolation \(L^p-L^q\) estimates (2.18, 2.20), and (5.6) to conclude
The next step is to estimate
where, by Hölder’s inequality, (2.21), and (2.15),
The last step consists in controlling the numerical viscosity. To this end, we first claim that (5.10) gives rise to
Next, we get
whence, by virtue of (6.1) combined with (2.4) and (2.15), we may infer that
Remark 6.1
Our estimates of the numerical viscosity are in fact considerably better than in [19, Section 5.3, Lemma 5.5]. This is due to the fact that the pressure considered here satisfies \(p'(0) > 0\) yielding (5.10).
Using the standard representation theorems for bounded linear forms on Sobolev spaces, we reformulate the continuity method as:
for any \(\phi \in C^\infty _c(R^3)\), where \(\mathbf{R}^1_h\) is a piecewise constant with respect to the time variable \(t \in [0,T]\) such that
6.2 Momentum method
In order to derive a consistency formulation of the momentum method, we take
as a test function in the momentum method (3.3). Note that, in accordance with the hypothesis (1.13), \(\phi \in C^\infty _c(\Omega _h;R^3)\) as soon as \(h > 0\) is small enough. By virtue of (2.7), (2.8), we have
and
Consequently, we may rewrite (3.3) in the form
Our goal is to estimate the four integrals on the right-hand side of (6.4). We proceed in several steps.
6.2.1 Error in the discretized time derivative
We have
where, by virtue of Hölder’s inequality and the estimate (2.9),
In accordance with the energy estimates (5.7), we have
Applying a similar treatment to the second integral we get
where the first integral on the right-hand side is controlled by means of (5.7).
Thus we may infer that
6.2.2 Error in the upwind term
Take \(F = \widehat{ \Pi ^V_h [ \phi ] } = \Pi ^Q_h \Pi ^V_h [\phi ]\) in (2.14) to obtain
Step 1 Applying Hölder’s inequality to \(I_1\) we obtain
where the first term is bounded in \(L^2(0,T)\) in view of the energy estimates (5.8).
Next, as \(\mathbf{u}^k_h\) are continuous on each element, we have
where, in accordance with the trace estimates (2.15, 2.16), and the \(L^p-L^q\) estimates (2.18),
Finally, by virtue of (2.4, 2.9),
As \(\gamma > 3\), we conclude that
Step 2 Next, we have
where, in accordance with (5.10), the first integral is uniformly bounded in \(L^2(0,T)\).
As for the second integral, we use Hölder’s inequality to deduce that
Next, by virtue of the trace estimate (2.15) and Hölder’s inequality,
Furthermore, by (2.16) and Hölder’s inequality,
where we have used (2.17).
Finally, using the time estimates (2.20) we infer that
Summarizing we conclude that
Step 3 Another application of Hölder’s inequality gives rise to
Now, by virtue of (2.17) and (2.9),
yielding the desired conclusion
Step 4 The last integral
can be handled in the same way as its counterpart in the continuity method.
6.2.3 Bounds on numerical dissipation
Finally, the numerical viscosity
can be estimated by means of (5.9, 5.10) in a similar way as in the continuity method
Summing up (6.7)–(6.10) we obtain the consistency formulation of the momentum method:
for any \(\phi \in C^\infty _c({\Omega }; R^3)\), where \(\mathbb {R}^2_h\) is piecewise constant in time,
7 Convergence of the numerical solutions
We are ready to establish convergence of solutions of our numerical method to a weak solution of the limit problem. We take advantage of the consistency formulation derived in the preceding section that converts the problem to the framework of the mathematical theory developed in [10] and [20]. The reader may also consult [1] for a complete existence proof based on the technique of time discretization very close to the numerical method applied in the present paper. Throughout the whole section we shall systematically use our convention that all quantities defined on \(\Omega _h\) are extended to be zero outside \(\Omega _h\).
7.1 Local pressure estimates
The uniform bound (5.4) is not sufficient for passing to the limit in the pressure \(p(\varrho )\), the latter being bounded only in the non-reflexive space \(L^\infty (0,T; L^1(R^3))\). To get better integrability of the pressure, we use the quantities
where
and \(\mathcal {F}\) denotes the standard Fourier transform, as test functions in the consistency formulation (6.11) of the momentum method:
Furthermore, using a discretized version of the integration by parts formula and the consistency formulation of the momentum method (6.2), we deduce that
We observe that the expression on the right-hand side of (7.1) is bounded uniformly for \(h \rightarrow 0\). Indeed combining the estimates (5.3), (5.4) we have
The integrals on the right-hand side of (7.1) can therefore be estimated in the same way as in [10, Chapter 5] and we may conclude that
7.2 Weak sequential compactness
In accordance with the uniform estimates (5.4)–(5.6), there is a subsequence of numerical solutions such that
and
Moreover, we have \(\varrho \ge 0\), and, by virtue of (5.1),
Next, it follows from (2.9) that
in particular,
provided \(\widehat{\mathbf{u}}_h\) is extended to be zero outside \(\Omega _h\).
Finally, we observe that (5.5) implies
whence the limit velocity field satisfies
Remark 7.1
The fact that the weak limit of \(\nabla _h\mathbf{u}_h\) coincides with \(\nabla _x\mathbf{u}\) follows from the “density” of the spaces \(V_{h,0}\) in \(W^{1,2}_0\) stated in (2.9).
In addition, we may use Lemma 2.1 to construct the smooth approximations \(R^V_h[ \mathbf{u}_h ]\),
It follows from (2.26) that the limit \(\mathbf{u}\) vanishes on any compact \(K_e \subset R^3 {\setminus } \overline{\Omega }\). Since \(\Omega \) is Lipschitz, we conclude that
Remark 7.2
Note that this is the only point, where certain regularity of \(\partial \Omega \) is needed. As already pointed out, the assumption that \(\Omega \) is Lipschitz can be considerably relaxed.
To establish the weak convergence of convective terms, we need the following result that can be seen as a variant of [18, Lemma 2.3].
Lemma 7.1
Let \(\{ v_h \}_{h > 0}\), \(\{ w_h \}_{h > 0}\) be two sequences of functions in \((0,T) \times Q\), Q a domain in \(R^N\), such that
Then
In agreement with the gradient estimates (5.5) and the compactness properties of the space \(H^1_{V_h}\) stated in (2.25), we observe that the sequence \(\{ \mathbf{u}_h \}_{h > 0}\) satisfies the hypothesis (7.12) with \(p_2 = q_2 = 2\;Q = \Omega \), while the hypothesis (7.11) can be checked for \(\varrho _h\), \(\varrho _h \widehat{\mathbf{u}}_h\) with the help of the consistency formulations (6.2, 6.11). Thus successive application of Lemma 7.1 gives rise to the following limits:
and
Remark 7.3
As for the exponent q in (7.14), we recall that
by interpolation.
7.3 Limit in the field equations
At this stage we are ready to pass to the limit in the consistency formulation of the numerical method. Letting \(h \rightarrow 0\) in (6.2, 6.11) we obtain
for any \(\varphi \in C^\infty _c([0,T) \times R^3)\);
for any \(\varphi \in C^\infty _c([0,T) \times {\Omega }; R^3)\).
Remark 7.4
In view of the local pressure estimate (7.5) we may assume that
7.4 Strong convergence of the density
In order to finish the proof of convergence we have to show a.e. pointwise convergence of the numerical densities in order to replace \(\overline{p(\varrho )}\) by \(p(\varrho )\) in (7.16). To this end, we use the method of Lions [20] based on a “weak continuity” property of the effective viscous flux. Going back to (7.1), (7.2), we focus on the term
Following [19], we perform integration by parts to obtain
where the error is estimated by means of [19, Lemma 8.2] as
Returning to (7.17), we get
with
where \(R^Q_h\) are the regularizing operators introduced in Lemma 2.1. Thus, recalling the bounds (5.10) and applying Lemma 2.1, we obtain
where \(K \subset \Omega \) is a compact set containing the spatial support of the function \(\eta \). In particular, the integral
vanishes for \(h \rightarrow 0\) and may be included in the error term on the right-hand side of (7.1).
Similarly, by the same token,
whence, in accordance with (7.18) we may replace
Summing up the previous estimates and regrouping terms in (7.1) we obtain
with the error term \(E_h (\varphi , \eta ) \rightarrow 0\) as \(h \rightarrow 0\) for any fixed \(\varphi , \eta \).
Remark 7.5
It is worth noting that this is the only step in the proof, where we have used the artificial viscosity term included in the upwinding.
Now we apply a similar treatment to the limit Eq. (7.16), specifically, we use the test functions
After a straightforward manipulation (cf. [10, Chapter 6]) we arrive at
The principal idea due to Lions [20] is that all terms on the right-hand side of (7.19) converge to their counterparts in (7.20). This has been proved in the continuous case in [20] and for the time discretization problem in [1, Section 3.3], Lions [20]. The same result at the level of numerical discretization was obtained by Karlsen and Karper [18], Karper [19]. Here, we recall that the error terms in (7.19) vanish for \(h \rightarrow 0\); whence the most difficult task is to show that
Moreover, in view of the numerical dissipation estimates (5.7), we may replace \((\varrho _h \widehat{\mathbf{u}}_h)(t - \Delta t)\) by \(\varrho _h \widehat{\mathbf{u}}_h\). Finally, we observe that the velocity field \(\mathbf{u}_h\) can be approximated by its spatial regularization in the spirit of Lemma 2.1,
In particular, we may write \(R^V_h[ \mathbf{u}_h ]\) in place of \(\mathbf{u}_h\) in (7.21). Now, the limit (7.21) can be verified exactly as in [1, Section 3.3] or Karper [19, Lemma 9.3].
Thus we get the desired conclusion - the effective viscous flux identity due to Lions [20]:
as \(h \rightarrow 0\) for any \(\varphi \in C^\infty _c((0,T) \times \Omega )\), which yields the crucial relation
The inequality (7.23) implies convergence \(\varrho _h \rightarrow \varrho \) a.e. in \((0,T) \times \Omega \). Indeed the regularization procedure of DiPerna and Lions [7] can be applied to show that \(\varrho \) is a renormalized solution of the continuity equation, in particular,
cf. [10, Chapter 6]. On the other hand, passing to the limit in the renormalized continuity method (4.1) for \(b(\varrho ) = \varrho \log (\varrho )\) we obtain
Combining (7.23)–(7.25) we get
yielding the desired conclusion
Seeing that the energy inequality (1.12) follows from (4.5) we have completed the proof of Theorem 3.1.
8 Unconditional convergence
Our ultimate goal is to discuss the situation when both the data \(\varrho _0\), \(\mathbf{u}_0\) and the underlying physical domain \(\Omega \) are regular. Specifically, we claim the following result concerning unconditional convergence of bounded numerical solutions.
Theorem 8.1
In addition to the hypotheses of Theorem 3.1, suppose that \(\Omega \) is a bounded domain of class \(C^{2 + \nu }\) and the initial data satisfy
and \(\eta = 0\). Moreover, suppose that there exists a positive constant r such that
Then the convergence claimed in Theorem 3.1 is unconditional, meaning the limit solution \(\varrho \), \(\mathbf{u}\) is regular, unique, and the whole family of numerical solutions converges to it.
Proof
The hypothesis (8.1) implies that the density component of the limit solution is bounded. Using the conditional regularity result proved in [11, Theorem 2.4] and [15, Theorem 4.6] we conclude that the limit solution is regular whence unique.\(\square \)
Under the condition described in Theorem 8.1, it is possible to obtain qualitative estimates on the rate of convergence of the numerical scheme, see [12].
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E. F. and M. M. acknowledges the support of the GAČR (Czech Science Foundation) project 13-00522S in the framework of RVO: 67985840.
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Feireisl, E., Karper, T. & Michálek, M. Convergence of a numerical method for the compressible Navier–Stokes system on general domains. Numer. Math. 134, 667–704 (2016). https://doi.org/10.1007/s00211-015-0786-6
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DOI: https://doi.org/10.1007/s00211-015-0786-6