Abstract
In this paper, we study a mixed discontinuous Galerkin-finite element method (DG-FEM) for solving the semi-stationary compressible Stokes system in a bounded domain. The approximation of continuity equation is obtained by a piecewise constant discontinuous Galerkin method. The discretization of momentum equation is obtained by conforming Bernardi–Raugel finite elements. The convergence of mixed DG-FEM for nonlinear, isentropic stokes problem is rigorously established by compactness arguments and the existence analysis of Lions on the discrete level. Employing the continuous relative energy functional method and a detailed consistency analysis, we derive two error estimates for the numerical solution of the semi-stationary isentropic stokes system. In particular, we establish the \(L^2\) error estimates for the pressure. All convergence results do not require the boundedness of numerical solutions.
Similar content being viewed by others
Avoid common mistakes on your manuscript.
1 Introduction
Let \(\varOmega \subset {\mathbb {R}}^d\), \(d=2,3\) be a bounded domain, we consider the following semi-stationary compressible Stokes problem:
where \(\rho \) is the fluid density and \(\varvec{u}\) is the velocity. The parameters coefficients \(\mu \) and \(\lambda \) are assumed to be constant and satisfy \(\mu >0\), \(d\lambda +2\mu >0\). The pressure \(p(\rho )\) is governed by the isentropic equation (or Boyle’s law):
where \(\gamma >1\) is the adiabatic exponent. The internal energy \({\mathcal {H}}\) is given by \({\mathcal {H}}(\rho )=\frac{p(\rho )}{\gamma -1}\). The system (1.1)–(1.2) is supplemented with initial conditions for the density
Together with the following no-slip boundary condition for the velocity
In recent years, numerical methods for compressible Stokes equations have received some attention. In the pioneering work of [20], the authors proposed a low order mixed finite element-finite volume (FE-FV) scheme based on nonconforming \(P_1\) (also called Crouzeix–Raviart) finite element for solving the stationary compressible isothermal Stokes problem and analyzed its convergence to a weak solution of the continuous problem. After that, the convergence of mixed FE-FV scheme to weak solution of the isentropic case under the assumption of \(\gamma >1\) has been established by Eymard et al. [10]. Meanwhile, they generalized the results to the well known Marker-and-Cell (MAC) scheme in [9]. Later, the convergence of mixed FE-FV scheme to weak solution of the general compressible Stokes problem (\(p=\varphi (\rho )\), where \(\varphi \) is a superlinear nondecreasing function from \({\mathbb {R}}\) to \({\mathbb {R}}\)) under the hypothesis \(\gamma >1\) was proved by Fettah and Gallouët in [18]. The models studied in the above mentioned literature are all steady state compressible stokes models. The semi-steady compressible Stokes model is known as a reasonable approximation of the isentropic Navier–Stokes equations when the convective effects can be neglected. The convergence of mixed DG-FEM based on nonconforming \(P_1\) finite element for the semi-steady compressible Stokes flow with a Navier boundary condition was shown by Karlsen and Karper in [26]. Meanwhile, they proposed and analyzed the convergence of a new mixed DG-FEM (here the velocity and vorticity were approximated by the div-conforming and curl-conforming Nédélec finite element spaces) to the semi-stationary compressible Stokes systems in [27]. We also mention that the convergence of the MAC scheme for the semi-stationary compressible Stokes flow with Dirichlet boundary conditions was proved in [21]. Very recently, a mixed FE-FV scheme based on Bernardi–Raugel finite element scheme for the stationary compressible isothermal Stokes system was proposed in [2]. The authors gave a convergence proof for the isothermal Stokes equations and investigated the convergence of numerical solutions to its incompressible limit. The convergence analysis is restricted to the isothermal Stokes equations (the pressure of the form (1.2) with \(\gamma =1\)) and the extension to the case \(\gamma >1\) remains open.
The aim of this paper is to show the convergence and error estimates of a mixed DG-FEM based on Bernardi–Raugel finite element for the semi-stationary (isentropic) compressible Stokes equations. This work consists of two major parts. The first part of this paper is to show the convergence of a mixed DG-FEM to a weak solution of the system (1.1) for any \(\gamma >1\). The convergence result of this paper is nontrivial compared to the existing literature. On the one hand, we see that the function \(\varvec{v}_h=\varPi _h^{{\mathbb {V}}}\nabla \varDelta ^{-1}[\rho _h]\) is not a solution to the div-curl problem
where \(\varPi _h^{{\mathbb {V}}}\) is the reconstruction interpolation operator of Bernardi–Raugel finite element space \({\mathbb {V}}_h\). Therefore, it is more difficult to obtain the discrete version of the effective viscous flux compared to [26], which will complicate the convergence analysis in this paper. On the other hand, the convergence analysis of this paper is valid for the semi-stationary (isentropic) compressible Stokes equations for any \(\gamma >1\). Of course, it is also valid for the stationary compressible Stokes equations with a slight modification, which fills the gap in the convergence analysis of [2] for the case \(\gamma >1\). We also want to remark that the \(\varvec{H}^1\)-conforming Bernardi–Raugel finite element has several advantages compared to the nonconforming Crouzeix–Raviart element used in the references [10, 18, 20, 23, 26]. Firstly, the conforming finite element method has less number of degrees of freedom which results in a cheaper computational cost. Secondly, the Korn’s inequality is admissible for the conforming method employed to approximate the velocity unknown. It is well known that the Korn’s inequality does not hold for the nonconforming Crouzeix–Raviart finite element space. Therefore, the conforming setting of this paper is easier to generalize to other viscous stress tensor compared to the nonconforming method in the references. Third, the convergence proofs of the conforming setting is less “structure dependent” than the nonconforming method. In other words, the methodology of the convergence proofs in this paper can be easily generalized to other numerical schemes.
The second part of this paper is to derive an error estimate between the mixed DG-FEM solution of the semi-stationary compressible Stokes system and its strong solution. By a detailed consistency analysis and the relative energy functional method introduced in reference [13], two error estimates for the numerical solutions of problem (1.1) under the hypothesis \(\gamma >\frac{6}{5}\) are proved in this paper. All the error results are unconditional in the sense that we do not require the boundedness of numerical solutions and the CFL like condition on the temporal mesh size. The relative energy method was originally designed to analyze the weak-strong uniqueness property of the compressible Navier–Stokes equations. Recently, this idea has been used to analyze the error estimate of numerical schemes of compressible Navier–Stokes system under the hypothesis \(\gamma >\frac{3}{2}\), such as the mixed DG-FEM based on nonconforming Crouzeix–Raviart finite element [12, 23], the implicit MAC scheme [24] and the finite difference method [31]. The error analysis of this paper uses similar analytical techniques but with some modifications. Firstly, our analysis is based on a detailed consistency analysis and the continuous relative energy functional method, rather than the discrete version used in the above literatures. Secondly, our numerical scheme is different from the above work and it requires to deal with some different technical estimates. Thirdly and more importantly, we derive the unconditional \(L^2\) error estimate of pressure under the assumption of \(\gamma >\frac{6}{5}\). To the best of our knowledge, this is the first unconditional error estimate of pressure for the compressible flows.
A brief overview of this work is provided as follows. In the next section, we introduce some notations and preliminary knowledge for this paper. In Sect. 3, we consider a mixed DG-FEM based on Bernardi–Raugel finite element for the semi-stationary compressible Stokes equations. After that, we deduce the discrete energy law, a priori estimate of pressure, the existence of numerical solutions and some uniform bounds. In Sect. 4, we establish the consistency formulation for the continuity equations. In Sect. 5, we show the boundedness of discrete time derivative and an important priori estimates for the density. The convergence of mixed DG-FEM for the nonlinear, isentropic Stokes equations is proved by compactness arguments and the existence analysis of Lions on the discrete level in Sect. 6. In Sect. 7, an unconditional error estimate for mixed DG-FEM solution of the problem (1.1) under the hypothesis \(\gamma >\frac{6}{5}\) is proved by the relative energy functional method.
2 Notation and Preliminaries
In this section, we introduce some notations and preliminary results used in this paper. For any \(1\le q\le \infty \), \(L^q(\varOmega )\) denotes the usual Lebesgue space on \(\varOmega \). For all non-negative integers k and r, \(W^{k,r}(\varOmega )\) stands for the standard Sobolev spaces. We write \(H^k(\varOmega )=W^{k,2}(\varOmega )\). We define \(H_0^1(\varOmega )\) as the subspace of \(H^1(\varOmega )\), which is zero on \(\partial \varOmega \). The vector-valued quantities will be denoted in boldface notations, such as \(\varvec{u}=(u_i)_{i=1}^d\) and \(\varvec{L}^2(\varOmega )=(L^2(\varOmega ))^d\).
Hypothesis 2.1
The initial data \(\rho _0\) satisfies the following properties:
Definition 2.1
We say that \((\rho ,\varvec{u})\) is a weak solution of the problem (1.1) if it satisfies the following properties:
(i) The solution \((\rho ,\varvec{u})\) satisfied the regularity requirements
(ii) For any test fuctions \((\varphi ,\varvec{v})\in C_0^\infty ((0,T)\times \varOmega )\times \varvec{C}_0^\infty ((0,T)\times \varOmega )\) and \(t_F\in [0,T]\), there holds the weak formulation
(iii) The solution \((\rho ,\varvec{u})\) satisfies the energy inequality
Next, we recall the following renormalized solution argument introduced by DiPerna and Lions (see e.g., [6]).
Definition 2.2
We say that \((\rho ,\varvec{u})\in L^\infty (0,T;L^\gamma (\varOmega ))\times L^2(0,T;\varvec{H}_0^1(\varOmega ))\) is a renormalized solution of the continuity equation \(\partial _t\rho +{\text {div}}(\rho \varvec{u})=0\) if the identity
in \({\mathcal {D}}'((0,T)\times \varOmega )\) holds for any \(\varPhi \in C[0,\infty )\cap C^1(0,\infty )\) with \(\varPhi (0)=0\), \(\varPsi (\rho )=\varPhi '(\rho )\rho -\varPhi (\rho )\) and \(\varPhi (\rho ),\varvec{u}\varPhi (\rho )\in L^1((0,T)\times \varOmega )\).
Finally, we recall the following well-known lemma [30] which says that the weak solution \(\rho \) is a renormalized solution.
Lemma 2.1
Suppose that couple \((\rho ,\varvec{u})\in L^2((0,T)\times \varOmega )\times L^2(0,T;\varvec{H}_0^1(\varOmega ))\) satisfies the continuity equation in the weak sense (2.1). Then \((\rho ,\varvec{u})\) is also renormalized solution according to Definition 2.2.
3 Numerical Method
In this section, we consider a mixed DG-FEM based on Bernardi–Raugel finite element for solving the compressible stokes problem (1.1).
3.1 Finite Dimensional Function Spaces
In order to introduce the mixed DG-FEM scheme, the mesh and some discrete function spaces are defined. Let \({\mathcal {T}}_h\) be a quasi-uniform tetrahedral partition of \(\varOmega \) with \({\overline{\varOmega }}=\cup _{K\in {\mathcal {T}}_h}{\overline{K}}\), \(K_i\cap K_j=\emptyset \) for \(K_i,K_j\in {\mathcal {T}}_h\), \(i\ne j\). The mesh size is defined by \(h=\max _{K\in {\mathcal {T}}_h}h_K\), where \(h_K\) is the mesh size of K. We write \({\mathcal {F}}_{h}\) as the set of faces in \({\mathcal {T}}_{h}\), while F is the face. Furthermore, \({\mathcal {F}}_{h,ext}\) is the set of faces \(F\in \partial \varOmega \), while \({\mathcal {F}}_{h,int}={\mathcal {F}}_h\setminus {\mathcal {F}}_{h,ext}\).
In order to discretize the problem, we introduce two families of finite-dimensional spaces. Before proceeding further, we write \({\mathbb {P}}_n(K)\) as the space of polynomials of degree n, while \({\mathbb {P}}_n^d(K)=[{\mathbb {P}}_n(K)]^d\). We define the space of piecewise constant functions
for the approximation of the density. In addition, we introduce the associated projection operator
By recalling the standard Poincaré and Jensen’s inequalities, we have the following interpolation error estimates
for any \(K\in {\mathcal {T}}_h\) and \(1\le q\le \infty \). We define the trace
where \(\varvec{n}_{F}\) is the outer normal vector to the face F. Moreover, we define the jumps \(\llbracket {v}\rrbracket :=v^{+}-v^{-}\) for any \(F\in {\mathcal {F}}_{h,int}\). Finally, we introduce the semi-norm of the space \({\mathbb {Q}}_h\)
We employ the Bernardi–Raugel finite element space (see, e.g., [3, 25])
for the approximation of the velocity. The local Bernardi–Raugel finite element space \({\mathbb{B}\mathbb{R}}(K)\) is given by
where \(\lambda _j\) is the barycentric coordinate of K and \(\varvec{n}_i\) is the unit outward normal to \(F_i\subset \partial K\). We introduce the reconstruction interpolation operator (see, e.g., [25, Chapter II])
The interpolation operator \(\varPi _{h}^{{\mathbb {V}}}\) has the following error estimates (see, e.g., [25, Chapter II, Lemma 2.2 and 2.8]):
where \(|\cdot |_{m,\varOmega }\) is the semi-norm of \(\varvec{H}^m(\varOmega )\) and \(m=0,1\), \(k=1,2\). Obviously, taking \(k=m=1\) in (3.2), the interpolation operators \(\varPi _h^{{\mathbb {V}}}\) have the following \(\varvec{H}^1\)-stable
Finally, we introduce some basic estimate for finite dimensional function spaces. By recalling the following inverse estimate from [5, Theorem 4.5.11], there holds
for any polynomial functions \(\varvec{v}|_K\in {\mathbb {P}}_n^d(K)\), \(K\in {\mathcal {T}}_h\), where \(C>0\) is a generic constant independent of the mesh-size h, m and r are two real numbers with \(0\le m\le r\), \(q_1\) and \(q_2\) are two integers with \(1\le q_1,q_2\le \infty \). By applying the scaling arguments and the trace theorem, we obtain
for any \(K\in {\mathcal {T}}_h\) and \(1\le q\le \infty \) and \(\varvec{v}\in \varvec{W}^{1,p}(K)\); see, e.g., [1]. Moreover, we apply the inverse estimate (3.4) and the trace inequality (3.5) to obtain
for any \(K\in {\mathcal {T}}_h\) and \(1\le q\le \infty \), \(\varvec{v}\in {\mathbb {P}}_n^d(K)\).
3.2 The Discretization of the Convection Term
Before introducing the scheme, we discuss the approximation of the convection operators in the continuity equation. To this end, we define the standard upwind operator \({\text {Up}}[r_h,\varvec{v}_h]\) on a face F, which is described by
where \([\varvec{v}_{h,F}\cdot \varvec{n}]^{+}:=\max \{0,\varvec{v}_{h,F}\cdot \varvec{n}\}\) and \([\varvec{v}_{h,F}\cdot \varvec{n}]^{-}:=\min \{0,\varvec{v}_{h,F}\cdot \varvec{n}\}\), \(\varvec{v}_{h,F}:=\frac{1}{|F|}\int _F{\varvec{v}_h}dS\). By applying the following lemma, we can show the distributional error of the convective term and its numerical analogue.
Lemma 3.1
For all \(r_h\in {\mathbb {Q}}_h\) and \(\varvec{v}_h\in {\mathbb {V}}_h\), \(\varphi \in H_0^1(\varOmega )\), we conclude that
Proof
By the same procedure as in [14, Section 2.3], we easily see that
for any \(r_h,g_h\in {\mathbb {Q}}_h\), \(\varvec{u}_h\in {\mathbb {V}}_h\) and \(\varphi \in H_0^1(\varOmega )\). It can easily be seen that
Combining the above analysis, the proof is thus complete. \(\square \)
3.3 Numerical Scheme
For the time discretization, let N be a fixed integer and \(0=t_0<t_1<\cdots <t_N=T\) be a uniform partition of [0, T] with time-step size \(\tau =T/N\). Moreover, let \(t_n=n\tau \) be the discrete time points and \(v^n\) is the approximation value of the function v at time \(t_n\) for \(0\le n \le N\). For convenience, we introduce \(d_t\varvec{v}^n=(\varvec{v}^n-\varvec{v}^{n-1})/\tau \) and \(D_t\varvec{v}(t)=(\varvec{v}(t)-\varvec{v}(t-\tau ))/\tau \).
We initialize the scheme \(\rho _h^0:=\varPi _h^{{\mathbb {Q}}}[\rho _0]\). For any \(1\le n\le N\), we compute \((\rho _h^n,\varvec{u}_h^n)\in {\mathbb {Q}}_h\times {\mathbb {V}}_h\) by the following numerical scheme
Scheme 1
Given \(\rho _h^{n-1}\in {\mathbb {Q}}_h\), for any \((\varphi _h,\varvec{v}_h)\in {\mathbb {Q}}_h\times {\mathbb {V}}_h\), find \((\rho _h^n,\varvec{u}_h^n)\in {\mathbb {Q}}_h\times {\mathbb {V}}_h\) such that
Remark 3.1
(i) Taking \(\varphi _h=1\) in the discrete continuity equation (3.7), we can show \(\int _{\varOmega }{\rho _h^n}dx=\int _{\varOmega }{\rho _h^{n-1}}dx\). In other words, we have immediately the scheme satisfying the conservation of mass. (ii) The stabilization term in the discrete continuity equation is useful in the convergence analysis. More specifically, it provides control over the discrete semi-norm of \(\rho _h\) by some (negative) power of the mesh size h. We remark that the artificial stabilization term in the convergence analysis of compressible flows is introduced by [10, 18, 20].
The renormalized continuity scheme can derived by the following lemma and the proof can be referred to [14, Section 4.1] for more details.
Lemma 3.2
(Renormalized continuity scheme). For any \(1\le n\le N\), let \((\rho _h^n,\varvec{u}_h^n)\in {\mathbb {Q}}_h\times {\mathbb {V}}_h\) satisfy the continuity scheme (3.7). Then \((\rho _h^n,\varvec{u}_h^n)\) also satisfies the following renormalized continuity scheme
for any \({\mathcal {B}}\in C^2({\mathbb {R}}_{+})\) and \(\varphi _h\in {\mathbb {Q}}_h\), where \(\xi _{\rho ,h}^n\in {\text {co}}\{\rho _h^{n-1},\rho _h^n\}\) on each element \(K\in {\mathcal {T}}_h\) and \({\bar{\eta }}_{\rho ,h}^n,\eta _{\rho ,h}^n\in {\text {co}}\{\rho _h^n,(\rho _h^n)^{+}\}\) on each face \(F\in {\mathcal {F}}_h\), where \({\text {co}}\{a,b\}=[\min \{a,b\},\max \{a,b\}]\).
In the upcoming analysis, the discrete density solution \(\rho _h\) is necessary for positive. For this purpose, we recall the following lemma (see, e.g., [22, 26, 28]).
Lemma 3.3
For any \(1\le n\le N\), we assume that \(\rho _h^{n-1}>0\) in \(\varOmega \) and \(\varvec{u}_h^n\in {\mathbb {V}}_h\) holds. Then the solution \(\rho _h^n\in {\mathbb {Q}}_h\) of the discontinuous Galerkin method (3.7) satisfies
3.4 A Priori Estimates
In this subsection, we establish some a priori estimates for the discrete solutions of the scheme (3.7)–(3.8), including the energy estimate and the uniformly boundedness of pressure in \(L^2((0,T)\times \varOmega )\).
Theorem 3.1
(Discrete energy law) For any \(1\le m\le N\), the solution \((\rho _h^n,\varvec{u}_h^n)\) of the scheme (3.7)–(3.8) satisfies the following discrete energy law
where the discerte energy \({\mathcal {J}}_h\) and the discrete dissipation \({\mathcal {D}}_h\) are defined by
and the numerical diffusion terms \({\mathcal {D}}_{i,h}^n\) are given by
Proof
Taking \(({\mathcal {B}},\varphi _h)=({\mathcal {H}},1)\) in the renormalized continuity scheme (3.9) and by applying \({\mathcal {H}}'(\rho )\rho -{\mathcal {H}}(\rho )=p(\rho )\), we can show
Let \(\varvec{v}_h=\varvec{u}_h\) in (3.8), we conclude that
Combining the above analysis implies
for any \(1\le n\le N\). Summing (3.11) with respect to n from \(n=1\) to \(n=m\), we obtain (3.10). The proof is thus complete. \(\square \)
In order to show the \(L^2(\varOmega )\) estimate of pressure, we introduce an inverse of the divergence operator \({\textbf{B}}\), which satisfies the following result (see [4] and [19, Chapter 3]).
Lemma 3.4
Let \(\varOmega \subset {\mathbb {R}}^d\), \(d=2,3\) be a bounded domain. There exists a linear operator \({\textbf{B}}\) enjoying the properties
Moverever, the linear operator \({\textbf{B}}\) satisfies the following estimate
Next, we prove the stability estimate for the discrete pressure.
Theorem 3.2
Suppose that Hypothesis 2.1 is satisfied. For any \(1\le m\le N\), then the pressure \(p(\rho _h)\) satisfies the following estimate
Proof
Let \(r_h^n:=p(\rho _h^n)-\frac{1}{|\varOmega |}\int _{\varOmega }{p(\rho _h^n)}dx\) for \(1\le n\le N\). Taking \(\varvec{v}_h^n=\varPi _h^{{\mathbb {V}}}{\textbf{B}}[r_h^n]\) in (3.8) and by the definition of \(\varPi _h^{{\mathbb {V}}}\) and \({\textbf{B}}\), we can show
By applying Hölder inequality, the estimates (3.3) and (3.12), we obtain
Combining the above analysis, by applying Young inequality, we have
for any \(1\le n\le N\).
Summing (3.14) with respect to n from \(n=1\) to \(n=m\) and applying the discrete energy estimate (3.10) implies
By applying Hypothesis 2.1 for the inequality (3.15), we have (3.13). The proof is thus complete. \(\square \)
3.5 Existence of Numerical Solution
By applying Schaeffer’s fixed point theorem, we can show the existence of numerical solutions for the scheme (3.7)–(3.8) in this subsection. Firstly, we recall Schaeffer’s fixed point theory (see, e.g., [8, Theorem 9.2.4]):
Lemma 3.5
Let \({\mathcal {L}}:D\rightarrow D\) be a continuous mapping defined on a finite dimensional normed vector space D. Suppose that the set
is non empty and bounded. Then there exists \(z\in D\) such that \(z={\mathcal {L}}(z)\).
Then we can prove an existence result of numerical solutions for the scheme (3.7)–(3.8).
Theorem 3.3
For any \(1\le n\le N\), let \((\rho _h^{n-1},\varvec{u}_h^{n-1})\in {\mathbb {Q}}_h\times {\mathbb {V}}_h\) and \(\rho _h^{n-1}>0\) be given. Then, for each fixed \(h,\tau >0\), the scheme (3.7)–(3.8) has at least one solution
The proof of Theorem 3.3 can be found in “Appendix A.1”.
3.6 Uniform Bounds
In this subsection, we deduce some priori estimates from the discrete energy law (3.10). To this end, we need to extend the definition of discrete solution for any \(t\le T\). We define the piecewise constant interpolations of \(\rho _h^n\) by
and the piecewise constant interpolations of \(\varvec{u}_h^n\) by
The following stable results are proved by the discrete energy law and the \(L^2\) estimate of pressure, which is crucial in both error estimates and convergence analysis.
Lemma 3.6
Suppose that Hypothesis 2.1 is satisfied. Then the family \((\rho _h,\varvec{u}_h)\) defined in (3.16)–(3.17) satisfies the following estimates:
Lemma 3.7
Suppose that Hypothesis 2.1 is satisfied. Then the family \((\rho _h,\varvec{u}_h)\) defined in (3.16)–(3.17) satisfies the following estimates:
Lemma 3.8
Suppose that Hypothesis 2.1 and \(\gamma \ge 2\) are satisfied. Then the family \((\rho _h,\varvec{u}_h)\) defined in (3.16)–(3.17) satisfies the following estimates:
Proof
Taking \(({\mathcal {B}}(\rho ),\varphi _h)=(\rho ^2,1)\) in renormalized continuity scheme (3.9) and summing this result with respect to n from \(n=1\) to \(n=N\), we obtain
By applying Hölder inequality and the embedding \(L^{2\gamma }\hookrightarrow L^4\) and \(L^\gamma \hookrightarrow L^2\) for \(\gamma \ge 2\), we conclude that
Combining the above analysis with Hypothesis 2.1 and Lemma 3.6, we have the required estimates, the proof is thus complete. \(\square \)
4 Consistency Formulation of the Continuity Scheme
In this section, we establish the consistency formulation for the discrete solution of the numerical scheme (3.7)–(3.8). In other words, the discrete solution asymptotically satisfies the weak formulation of continuous problem.
Lemma 4.1
The family \((\rho _h,\varvec{u}_h)\) defined in (3.16)–(3.17) satisfies the following consistency formulation
for any \(\varphi \in L^2(0,T;H^1(\varOmega ))\), where the remainder functional \({\mathscr {R}}_h\) is given by
Proof
Taking \(\varphi _h=\varPi _h^{{\mathbb {Q}}}[\varphi ]\) in the continuity method (3.7) and summing this identity with respect to n from \(n=1\) to \(n=N\), we can show
It is easy to check that
By taking \((r_h,\varvec{v}_h)=(\rho _h^n,\varvec{u}_h^n)\) in Lemma 3.1 and summing this idenity with respect to n from \(n=1\) to \(n=N\), we conclude that
Combining the above analysis, we obtain (4.1). The proof is thus complete. \(\square \)
Next, the error estimate of the remainder term \({\mathscr {R}}_h\) of Lemma 4.1 is proved in the following lemma.
Lemma 4.2
Suppose that Hypothesis 2.1 is satisfied. There exists a constant \(C>0\) independent of h and \(\tau \), such that the error functional \({\mathscr {R}}_h\) of Lemma 4.1 satisfies the following estimates
where the parameters A and \(m_1\) are given by
Proof
We show the proof of this Lemma in four steps.
Bound on \({\mathcal {P}}_1\). We estimate this term for \(1<\gamma \le 2\) and \(\gamma >2\) separately. If \(1<\gamma \le 2\), by applying Cauchy–Schwarz inequality, we can show
where \({\mathcal {P}}_{1,1,1}\) and \({\mathcal {P}}_{1,1,2}\) are given by
It is easy to check that
For the term \({\mathcal {P}}_{1,1,2}\), by applying the inequality (4.5), we obtain
where \({\mathcal {P}}_{1,1,2,1}\) and \({\mathcal {P}}_{1,1,2,2}\) are defined by
Using the trace inequalities (3.5)–(3.6), we conclude that
Therefore, using the Hölder inequality and the interpolation error estimate (3.1), we get that
By a similar proof to the error estimate of \({\mathcal {P}}_{1,1,2,2}\), we find
Inserting (4.7) and (4.8) into (4.6), using Lemma 3.6, we have arrived at
By applying Lemma 3.7 and (4.9) to (4.4) leads to the bound
For the case \(\gamma >2\), by using Cauchy–Schwarz inequality, we obtain
where \({\mathcal {P}}_{1,2,1}\) is defined by
By virtue of the inequality (4.8), Lemmas 3.6 and 3.8, we have
Bound on \({\mathcal {P}}_2(\varphi )\). By applying the Hölder inequality, the inverse estimate (3.4) and the embedding \(L^{2\gamma }\hookrightarrow L^2\) for \(\gamma >1\), we obtain
Bound on \({\mathcal {P}}_3(\varphi )\). We shall treat the case \(1<\gamma \le 2\) and \(\gamma >2\) separately. If \(1<\gamma \le 2\), by applying the Cauchy Schwarz inequality, we have
where \({\mathcal {P}}_{3,1}\) and \({\mathcal {P}}_{3,2}\) are defined by
By employing the inequality (4.5), we can show
where \({\mathcal {P}}_{3,1,2,1}\) and \({\mathcal {P}}_{3,1,2,2}\) are given by
By applying the trace estimate (3.5)–(3.6) and the Poincaré inequality (3.1), the embedding \(L^{2\gamma }\hookrightarrow L^2\) for \(\gamma >1\), we get
Inserting (4.12) and (4.13) into (4.11), using Lemma 3.6, we conclude that
For the inequality (4.10), by using (4.14) and Lemma 3.7, we easily see that
For the case \(\gamma >2\), by applying the Cauchy Schwarz inequality, we obtain
where \({\mathcal {P}}_{3,2,1}\) is given by
According to Lemma 3.8 and the estimate (4.12), we have
Bound on \({\mathcal {P}}_4(\varphi )\). By employing the Hölder inequality, the trace estimates (3.5)–(3.6), the Poincaré and inverse inequalities, the embedding \(L^{2\gamma }\hookrightarrow L^2\) for \(\gamma >1\), we easily establish that
Combining the above analysis, we have the required estimate (4.3). The proof is thus complete. \(\square \)
5 Basic Estimates
This section establishes the boundedness of discrete time derivative \(D_t\rho _h\) and a priori estimate of discrete density \(\rho _h\) in \(L^2(0,T;{\mathbb {Q}}_h)\).
Lemma 5.1
Suppose that the conditions of Lemma 4.2 are satisfied, then the discrete time derivative \(D_t\rho _h\) satisfies
Proof
Let \(\phi \in L^{2m_1}(0,T;W^{1,6}(\varOmega ))\) such that \(\Vert \phi \Vert _{L^{2m_1}(0,T;W^{1,6}(\varOmega ))}=1\). Taking \(\varphi _h=\varPi _h^{{\mathbb {Q}}}[\phi ]\) in (3.7) and summing this result with respect to n from \(n=1\) to \(n=N\), applying the same argument as Lemma 4.1, we infer that
Using Hölder inequality, Lemmas 4.2 and 3.6, we conclude that
This inequality immediately implies Lemma 5.1. The proof is thus complete. \(\square \)
Lemma 5.2
Suppose that Hypothesis 2.1 and the CFL condition \(\tau \approx h\) are satisfied, there exists \(\epsilon _0>0\) and \(0<\delta <1\) such that for any \(0<\epsilon <\epsilon _0\),
Proof
We divide our proof in two steps. Firstly, if \(1<\gamma \le 2\), by applying Cauchy–Schwarz inequality, we obtain
where \({\mathcal {P}}_5\) and \({\mathcal {P}}_6\) are given by
According to the trace estimate (3.6) and the inequality (4.5), we infer that
On the one hand, for \(1<\gamma <\frac{4}{3}\), it is easy check that \(4-\gamma >2\gamma \). Therefore, by applying the inverse estimate (3.4) and the CFL condition \(\tau \approx h\), we have
On the other hand, for the case \(\frac{4}{3}\le \gamma <2\), by using the embedding result \(L^{2\gamma }\hookrightarrow L^{4-\gamma }\), we conclude that
Inserting (5.5) and (5.6) into (5.4), using Lemma 3.6, we obtain
By substituting (5.7) into (5.3), using Lemma 3.7, we get that
Secondly, for the case \(\gamma >2\), by using Lemma 3.8, we easily see that
Combining the inequalities (5.8) and (5.9), we have the required estimate (5.2), where the parameter \(\alpha _0\) and \(\delta \) are given by
It is easy check that \(\delta <1\). The proof is thus complete. \(\square \)
Remark 5.1
(i) In fact, for the case of \(\gamma \ge \frac{4}{3}\), the CFL condition \(\tau \approx h\) is not required for the estimate (5.2). (ii) Lemma 5.2 plays a key role in deriving the discrete version of the effective viscous flux identity. See Lemma 6.7 and Theorem 6.2 for more on why it is needed.
6 Convergence Analysis
In this section, we will prove the family \((\rho _h,\varvec{u}_h,p(\rho _h))\) defined in (3.16)–(3.17) converges to weak solution ( see Definition 2.1). For that purpose, we first need to establish a spatial compactness estimate for Bernardi–Raugel finite element space.
Theorem 6.1
Let q satisfies \(2\le q<6\) and \(\frac{1}{q}=\frac{\theta }{2}+\frac{1-\theta }{6}\), \(\theta \in [0,1]\). For any \(\varvec{v}_h\in {\mathbb {V}}_h\), there exists a constant \(C>0\) such that the following estimate holds
The proof of Theorem 6.1 can been found in “Appendix A.2”.
According to Lemma 3.6 and Theorem 3.2, we can assert the existence of functions
such that the family \((\rho _h,\varvec{u}_h)\) defined in (3.16)–(3.17) exists suitable subsequences satisfy
The following lemma can be found in [26, Lemma 2.3].
Lemma 6.1
Let \(\{f_h\}_{h>1}^\infty \) and \(\{g_h\}_{h>1}^\infty \) be two function sequences on \((0,T)\times \varOmega \) such that
-
(i)
\(f_h\) and \(g_h\) converge weakly to f and g respectively in \(L^{p_1}(0,T;L^{q_1}(\varOmega ))\) and \(L^{p_2}(0,T;L^{q_2}(\varOmega ))\), where \(1\le p_1,q_1\le \infty \), \(\frac{1}{p_1}+\frac{1}{p_2}=\frac{1}{q_1}+\frac{1}{q_2}=1\).
-
(ii)
Assume that \(\frac{g_h(t,\varvec{x})-g_h(t-h,\varvec{x})}{h}\) is bounded in \(L^1(0,T;W^{-m,1}(\varOmega ))\), for some \(m\ge 0\) independent of h. And \(\Vert f_h(t,\varvec{x})-f_h(t,\varvec{x}-\varvec{\xi })\Vert _{L^{p_1}(0,T;L^{p_2}(\varOmega ))}\rightarrow 0\) as \(|\varvec{\xi }|\rightarrow 0\) uniformly in h.
Then, \(f_hg_h\) converges to fg in the sense of distributions on \((0,T)\times \varOmega \).
Next, we present a weak convergent results for \(\rho _h\varvec{u}_h\).
Lemma 6.2
Suppose that the condition of Lemma 4.2 are satisfied, then the family \((\rho _h,\varvec{u}_h)\) defined in (3.16)–(3.17) satisfies
Proof
From Lemma 5.1, we can show
By applying Theorem 6.1 and Lemma 3.6, we conclude that
By substituting (6.1)–(6.3) into Lemma 6.1, the proof is thus complete. \(\square \)
6.1 Limit in the Compressible Stokes Equations
In this subsection, we can show the limit \((\rho ,\varvec{u},p)\) constructed in (6.1) is a weak solution of Definition 2.1. The remaining major difficulty is to prove the pressure \(p(\rho _h)\rightarrow p(\rho )\).
Lemma 6.3
Suppose that the condition of Lemma 4.2 is satisfied, then the accumulation point \((\rho ,\varvec{u})\) constructed in (6.1) satisfies the weak formulation (2.1).
Proof
We pass to the limit with \(h,\tau \rightarrow 0\) in the consistency formulation (4.1). Firstly, we rewrite the discrete time derivative term
where \(t_{\dag }\in (t,t+\tau )\) and \(t_{\ddag }\in (0,\tau )\). By applying Lemma 3.6 and the embedding \(L^\gamma \hookrightarrow L^1\), we have
According to (6.1) and \(\varPi _h^{{\mathbb {Q}}}[\rho _0]\rightharpoonup \rho _0\) in \(L^\gamma (\varOmega )\), we obtain
Next, by applying Lemma 6.2, we can show
Finally, by employing the inequality (4.3) of Lemma 4.2, we conclude that
Combining the above analysis, the proof is thus complete. \(\square \)
Lemma 6.4
Suppose that Hypothesis 2.1 is satisfied, the accumulation limit \((\rho ,\varvec{u})\) constructed in (6.1) satisfies the following weak formulation:
Proof
We define \({\mathcal {F}}_{{\mathbb {V}}_h}\) as the \(\varvec{L}^2\)-orthogonal projection operator from \(\varvec{L}^2(\varOmega )\) into \({\mathbb {V}}_h\). For any \(\varvec{v}\in \varvec{C}_0^\infty ((0,T)\times \varOmega )\), we can choose \(\varvec{v}_h={\mathcal {F}}_{{\mathbb {V}}_h}\varvec{v}\) and \(\varvec{v}_h^n=\frac{1}{\tau }\int _{t_{n-1}}^{t_n}{\varvec{v}_h}dt\) such that
for any \(t\in (0,T)\). Taking \(\varvec{v}_h^n\) in (3.8), multiplying by \(\tau \) and summing the results with respect to n from \(n=1\) to \(n=N\), we conclude that
Obviously, by applying (6.1) and (6.5), we have the required weak formulation (6.4). The proof is thus complete. \(\square \)
6.2 Strong Convergence of the Density
The strong convergence of the density is proved by the discrete version of the weak continuity property of the effective viscous flux introduced on the continuous level in [30]. For this purpose, we first introduce the following notation
where \(\varvec{v}\) is a vector-valued function. Obviously, if \(\varvec{v}\in \varvec{H}^1(\varOmega )\) and \(\varvec{w}\in \varvec{H}_0^1(\varOmega )\), we can show
Next, we report the following Lemma, which plays a key role in deriving the discrete version of the effective viscous flux.
Lemma 6.5
Let \(\varOmega \subset {\mathbb {R}}^d\) (\(d=2,3\)) be a bounded open set. For any \(1<r<\infty \) and \(q\in L^r(\varOmega )\), there exists \(\varvec{w}\in \varvec{W}^{1,r}(\varOmega )\) such that
where C only depends on \(\varOmega \) and r. Moreover, if \(q\in W^{1,r}(\varOmega )\) (or \(q\in W^{-1,r}(\varOmega )\)), it is possible to have \(\varvec{w}\in \varvec{W}^{2,r}(\varOmega )\) (or \(\varvec{w}\in \varvec{L}^r(\varOmega )\)) such that
Proof
It is easy to check that \(\nabla \varDelta ^{-1}[q]\) can be served as the desired solution, where \(\varDelta ^{-1}\) is the inverse of the Laplacian on \({\mathbb {R}}^3\), and here we applied to q extended by 0 outside \(\varOmega \). Obviously, \(\nabla \varDelta ^{-1}\) is a continuous linear operator from \(L^r(\varOmega )\) to \(\varvec{W}^{1,r}(\varOmega )\) and from \(W^{1,r}(\varOmega )\) to \(\varvec{W}^{2,r}(\varOmega )\), from \(W^{-1,r}(\varOmega )\) to \(\varvec{L}^r(\varOmega )\) (see e.g., [29, Lemma 8.3]). The proof is thus complete. \(\square \)
In the next step, we introduce the operator \(\varPi _h^{{\mathbb {Y}}}:{\mathbb {Q}}_h\mapsto {\mathbb {Y}}_h\) which interpolates the piecewise constant functions to the space of continuous finite element space \({\mathbb {Y}}_h\),
for any vertices A in the discretization, where \(N_A\) is the set of elements \(K\in {\mathcal {T}}_h\) of which takes A as its vertices. The operator \(\varPi _h^{{\mathbb {Y}}}\) satisfies the following results (see e.g., [10, Lemma 5.8]).
Lemma 6.6
For any \(q_h\in {\mathbb {Q}}_h\), there exists a constant \(C>0\), depending only on the shape-regularity of \({\mathcal {T}}_h\) such that
Then we can prove the following estimates.
Lemma 6.7
Suppose that the condition of Lemma 5.2 is satisfied, there exists a constant \(C>0\) such that the following estimates hold
Proof
By applying the inequality (5.2) and Lemma 6.6, we can show
According to the embedding \(L^{2\gamma }\hookrightarrow L^2\) for \(\gamma >1\), we have
These inequalities immediately implies
Combining the above analysis, the proof is thus complete. \(\square \)
Theorem 6.2
Suppose that the condition of Lemma 5.2 is satisfied. The family \((\rho _h,\varvec{u}_h)\) defined in (3.16)–(3.17) and the accumulation limit \((\rho ,\varvec{u})\) constructed in (6.1) satisfy the following convergence properties:
for any \(\psi \in C_0^\infty ((0,T))\) and \(\varphi \in C_0^\infty (\varOmega )\).
Proof
According to Lemmas 6.5, 6.7 and 3.6, the inequality (5.1), and there exists \(\varvec{w}_{{\mathbb {Y}},h}\in L^2(0,T;\varvec{H}^2(\varOmega ))\) and \(\varvec{w}_h\in L^2(0,T;\varvec{H}^1(\varOmega ))\) such that
and a generic constant C independent of h and \(\tau \) such that
Subtracting the right side of (6.8) from its left side, we obtain
By employing Lemmas 6.5 and 6.7, we can show
Using the Lemma A.4 together with the estimates (6.9), we deduce for a suitable subsequence that
In addition, the accumulation limit \(\varvec{w}\) satisfies the following properties
Taking \(\varvec{v}_h=\varPi _h^{{\mathbb {V}}}[\varphi \varvec{w}_{{\mathbb {Y}},h}]\) in (3.8), multiplying by \(\psi \in C_0^\infty ((0,T))\) and integrating from \(t=0\) to T, we derive
where \({\mathscr {R}}_{1,h}\) is given by
By applying the inequalities (3.2) and (6.9), we can show
Obviously, we have \({\text {div}}(\varphi \varvec{w}_{{\mathbb {Y}},h})=\varPi _h^{{\mathbb {Y}}}[\rho _h]\varphi +\varvec{w}_{{\mathbb {Y}},h}\cdot \nabla \varphi \) and \({\text {curl}}(\varphi \varvec{w}_{{\mathbb {Y}},h})=J(\varphi )\varvec{w}_{{\mathbb {Y}},h}\), where \(J(\varphi )\) is a matrix with entries involving some first-order derivatives of \(\varphi \). Combining the identities (6.6) and (6.13), we obtain
where \({\mathscr {R}}_{2,h}\) is defined by
Applying the Hölder inequality and Lemma 6.7 implies
The identity (6.15) can be rewritten as
where \({\mathscr {R}}_{3,h}\) is given by
Using the Hölder inequality and the estimate (6.10), we conclude that
Passing to the limit with \(h,\tau \rightarrow 0\) in (6.17), using (6.1) and (6.11), we find
By applying Lemma 3.6, the estimates (6.14), (6.16), (6.18) and \(\delta <1\), we get that
Taking \(\varvec{v}=\psi \varphi \varvec{w}\) in (6.4) and using the identity (6.12), imply
Combining the identities (6.19)–(6.21), we have the required discrete effective viscous flux identity (6.7). The proof is thus complete. \(\square \)
Lemma 6.8
(Strong convergence of \(\rho _h\)) Suppose that the condition of Lemma 5.2 is satisfied, then, passing to a subsequence if necessary
Proof
Firstly, we can show the sequences \(p(\rho _h)\rho _h\), \(\log (\rho _h)\rho _h\) and \(\rho _h{\text {div}}\varvec{u}_h\) have the following convergent properties:
in a suitable \(L^q((0,T)\times \varOmega )\) space with \(q>1\), where the overbar is used to denote the weak limit of a nonlinear function. According to the notation introduced above, we write \(p=\overline{p(\rho )}\), then it can be easily checked
for any \(\psi \in C_0^\infty (0,T)\) and \(\phi \in C_0^\infty (\varOmega )\). By applying the discrete effective viscous flux identity (6.7) and the identity (6.23), we conclude that
Take the following functions sequence \(\psi _m\in C_0^\infty ((0,T))\) and \(\phi _n\in C_0^\infty (\varOmega )\) such that
Let \((\psi ,\phi )=(\psi _m,\phi _n)\) in (6.24) and \(m,n\rightarrow +\infty \), by applying Lebesgue’s dominated convergence theorem, we obtain
For the identity (6.25), by employing Lemma A.1, we get that
According to Lemmas 2.1 and 6.3, we obtain \((\rho ,\varvec{u})\) is a renormalized solution of the continuity equation (2.1). Therefore, taking \(\varPhi (\rho )=\rho \log (\rho )\) in Definition 2.2 and integrating over \([0,t_F]\times \varOmega \) for the results, we can show
for any \(t_F\in [0,T]\).
Taking \(({\mathcal {B}}(\rho ),\varphi _h)=(\rho \log (\rho ),1)\) in the discrete renormalized continuity scheme (3.9) and passing to the limit with \(h,\tau \rightarrow 0\), we have
for any \(t_F\in [0,T]\). Subtracting the identity (6.27) from the inequality (6.28), we can show
for any \(t_F\in [0,T]\). Inserting (6.26) into (6.29), we obtain
On the other hand, according to Lemma A.2, we have
Combining the inequalities (6.30) and (6.31) implies
By applying Lemma A.3, we have the required result (6.22). The proof is thus complete. \(\square \)
Theorem 6.3
Suppose that the condition of Lemma 5.2 is satisfied. For any \(q_1\in [1,2\gamma )\) and \(q_2\in [1,2)\), then, passing to a subsequence if necessary
Proof
By applying (6.22) and Lemma 3.6, we have
Noticing \(x^{\gamma }\) and \(x^{\frac{1}{\gamma }}\) are increasing functions for \(x\in {\mathbb {R}}_+\) and \(\left( x-y\right) ^\vartheta \le x^\vartheta -y^\vartheta \) for \(x\ge y\ge 0\) and \(\vartheta >0\), we obtain
By employing the Hölder inequality, (6.32) and Lemma 3.6, we can show
Inserting (6.33) into (6.34), using (6.32), we can show
which implies that
By applying (6.35) and Lemma 3.6, we conclude that
where \(q_2\in [1,2)\). The proof is thus complete. \(\square \)
Combining Lemmas 6.3 and 6.4, and Theorem 6.3, we can obtain the main result of the first part of this paper:
Theorem 6.4
Let \(\varOmega \subset {\mathbb {R}}^d\), \(d=2,3\) be a bounded domain and assume that the viscosity coefficients \(\mu \) and \(\lambda \) satisfy \(\mu >0\) and \(d\lambda +2\mu >0\). Suppose that the pressure \(p=p(\rho )\) satisfies the assumption (1.2) with \(\gamma >1\). Furthermore, the initial values \(\rho _0\) satisfies Hypothesis 2.1. The family \((\rho _h,\varvec{u}_h)\) defined in (3.16)–(3.17) satisfies \(\rho _h>0\) for any \(h,\tau >0\) with \(\tau \approx h\) and \(0<\epsilon <\epsilon _0\). Then we have the following convergent properties:
for any \(1\le q_1<2\gamma \) and \(1\le q_2<2\), where \((\rho ,\varvec{u})\) is a weak solution of the semi-stationary compressible Stokes equations (1.1)–(1.4) in the sense of Definition 2.1.
Remark 6.1
(i) Theorem 6.4 provides an alternative proof of existence of weak solutions via a mixed DG-FEM based on Bernardi–Raugel finite element for the problem (1.1) under the hypothesis \(\gamma >1\). (ii) In the case \(\gamma >\frac{4}{3}\), the CFL condition \(\tau \approx h\) is not required for Theorem 6.4. It is worth noting that the values of adiabatic exponent \(\gamma \) in the convergence result without the CFL condition includes the real fluid range of \(\gamma \in [\frac{4}{3},\frac{5}{3}]\), such as the monoatomic gas (\(\gamma \sim \frac{5}{3}\)) and the diatomic gas (\(\gamma \sim \frac{7}{5}\)). (iii) Theorem 6.4 is also true with the external force \(\varvec{f}\ne \varvec{0}\in \varvec{L}^2((0,T)\times \varOmega )\) in the momentum equation.
7 Error Estimate
An unconditional error estimate for the semi-stationary compressible Stokes equations is established in the section. Note that the existence of weak solution to this model under the assumption of \(\gamma >1\) is proved by Theorem 6.4. Now we report the weak-strong uniqueness for this model. To this end, we introduce the following functional \({\mathbb {E}}:[0,\infty )\times (0,\infty )\rightarrow {\mathbb {R}}\), which is given by
Noticing that the function \({\mathcal {H}}\) is strictly convex in \((0,\infty )\), we obtain
Furthermore, the functional \({\mathbb {E}}(\rho \,|\,{\underline{\rho }})\) satisfies the following estimates (see, e.g., [13, 17] for more details)
where \(C({\underline{\rho }})\) is uniformly bounded if \({\underline{\rho }}\) lies in some compact subset of \((0,\infty )\). Finally, we introduce the relative energy functional of the problem (1.1), which is defined by
Theorem 7.1
Let \(\varOmega \subset {\mathbb {R}}^3\) be a bounded domain and assume that the viscosity coefficient \(\mu ,\lambda \) satisfies \(\mu >0\) and \(3\lambda +2\mu >0\). Suppose that the pressure \(p=p(\rho )\) satisfies the assumption (1.2) with \(\gamma >1\). Let \((\rho ,\varvec{u})\) be a weak solution to the problem (1.1) emanating from the initial data \((\rho _0,\varvec{u}_0)\) with the finite energy \(E_0:=\int _{\varOmega }{{\mathcal {H}}(\rho _0)}dx\) and finite mass \(M_0:=\int _{\varOmega }{\rho _0}dx\). Let \(({\underline{\rho }},\underline{\varvec{u}})\) be a strong solution of the same problem belonging to the class
emanating from the same initial data. Then
The proof of Theorem 7.1 can been found in [13, Theorem 4.1].
Next, we deduce the discrete version of the relative energy inequality from the scheme (3.7)–(3.8), which will play a key role in the subsequent error estimate. To this end, we first introduce the convenient notations
where \(({\underline{\rho }},\underline{\varvec{u}})\) is a strong solution of the problem (1.1) belonging to the class of \(C^2\) functions such that \(\underline{\varvec{u}}|_{(0,T)\times \partial \varOmega }=\varvec{0}\) and \(0<{\underline{\rho }}_{\min }\le {\underline{\rho }}\le {\underline{\rho }}_{\max }\). Furthermore, we define the piecewise constant temporal interpolations of \(({\underline{\rho }}_h^n,\underline{\varvec{u}}_h^n,{\underline{\rho }}^n,\underline{\varvec{u}}^n)\), \(1\le n\le N\), i.e., for any \(t\in [t_{n-1},t_n]\)
Theorem 7.2
Suppose that Hypothesis 2.1 and \(\gamma >1\) are satisfied. The family \((\rho _h,\varvec{u}_h)\) and \(({\underline{\rho }}_{\tau },\underline{\varvec{u}}_{\tau })\) are defined in (3.16)–(3.17) and (7.3), respectively. Then there exists a constant \(C>0\) independent of h and \(\tau \) such that
Proof
Using the identity \({\mathcal {H}}'(\rho )\rho -{\mathcal {H}}(\rho )=p(\rho )\) and the Hölder inequality, we can show
By employing Lemma 3.6, we have the estimate (7.4). This proof is thus complete. \(\square \)
Now we establish the discrete version of the relative energy inequality.
Theorem 7.3
Let the families \((\rho _h,\varvec{u}_h)\) and \(({\underline{\rho }}_{\tau },\underline{\varvec{u}}_{\tau })\) be defined as in (3.16)–(3.17) and (7.3), respectively. Then the discrete relative energy inequality holds, for any \(1\le m\le N\),
where the remainder terms \({\mathcal {R}}_i\) (\(1\le i\le 5\)) are defined by
Proof
First, taking \(\varvec{v}_h=\underline{\varvec{u}}_h^n\) in the discrete momentum equation (3.8), and summing this result with respect to n from \(n=1\) to \(n=m\), we conclude that
Next, using the same argument as Lemma 4.1 by taking \(\varphi ={\mathcal {H}}'({\underline{\rho }}_{\tau }^{\star })\) in Lemma 4.1, we obtain
Note that the numerical diffusion terms \({\mathcal {D}}_{i,h}^n\) (\(1\le i\le 3\)) in the discrete energy identity (3.10) are all positive, we have
By applying the identity
we rewrite
which implies that
According to the convexity of the function \({\mathcal {H}}\), we obtain
By using the inequality (7.10), we have
Combining the inequalities (7.6)–(7.9) and (7.11), we obtain the inequality (7.5). This proof is thus complete. \(\square \)
In the next step, we deduce the approximate version of the relative energy inequality from the estimate (7.5).
Theorem 7.4
Suppose that Hypothesis 2.1 is satisfied and the pressure \(p=p(\rho )\) satisfies the hypothesis (1.2) with \(\gamma >1\). Let the internal energy \({\mathcal {H}}\) be given by \({\mathcal {H}}(\rho )=\frac{p(\rho )}{\gamma -1}\). Let the families \((\rho _h,\varvec{u}_h)\) and \(({\underline{\rho }}_{\tau },\underline{\varvec{u}}_{\tau })\) be defined as in (3.16)–(3.17) and (7.3), respectively. Then there exists
such that for any \(1\le m\le N\), we have the approximate relative energy inequality holds,
where the remainder terms \({\mathcal {L}}_i\) (\(1\le i\le 3\)) are defined by
Proof
We start the proof from the discrete version of the relative energy inequality (7.5) derived in the previous Theorem 7.3. The terms \({\mathcal {R}}_i\) (\(i=2,4,5,6\)) will be transformed to a more convenient form, and the other terms \({\mathcal {R}}_i\) (\(i=1,3\)) will remain unchanged.
-
The term \({\mathcal {R}}_2\). By applying the Cauchy–Schwarz inequality and the estimate (3.2), we can show
$$\begin{aligned} |{\mathcal {R}}_2|\le&C\Vert \varvec{u}_h\Vert _{L^2(0,T;\varvec{H}^1(\varOmega ))}\Vert \underline{\varvec{u}}_h-\underline{\varvec{u}}_{\tau }\Vert _{L^2(0,T;\varvec{H}^1(\varOmega ))}\\ \le&C(E_0,\Vert \underline{\varvec{u}}\Vert _{L^\infty (0,T;\varvec{H}^2(\varOmega ))})h. \end{aligned}$$ -
The term \({\mathcal {R}}_4\). Firstly, by applying the Taylor formula, we have
$$\begin{aligned} {\mathcal {H}}'({\underline{\rho }}^n)-{\mathcal {H}}'({\underline{\rho }}^{n-1})={\mathcal {H}}''({\underline{\rho }}^n)({\underline{\rho }}^n-{\underline{\rho }}^{n-1})-\frac{1}{2}{\mathcal {H}}'''(\xi _{{\underline{\rho }}}^n)({\underline{\rho }}^n-{\underline{\rho }}^{n-1})^2, \end{aligned}$$(7.13)where \(\xi _{{\underline{\rho }}}^n\in {\text {co}}\{{\underline{\rho }}^n,{\underline{\rho }}^{n-1}\}\). Let \(\xi _{{\underline{\rho }}}(t,\cdot ):=\xi _{{\underline{\rho }}}^n\) for \(t\in [t_{n-1},t_n]\). By applying the identity (7.13), the term \({\mathcal {R}}_4\) can be rewritten as
$$\begin{aligned} {\mathcal {R}}_4={\mathcal {L}}_1+{\mathcal {L}}_{3,1}+{\mathcal {L}}_{3,2}, \end{aligned}$$where the remainder terms \({\mathcal {L}}_{3,i}\) are given by
$$\begin{aligned} {\mathcal {L}}_{3,1}:=&\int _0^{t_m}{\int _{\varOmega }{({\underline{\rho }}_{\tau }-\rho _h)\frac{p'({\underline{\rho }}_{\tau })}{{\underline{\rho }}_{\tau }}(D_t{\underline{\rho }}_{\tau }-[\partial _t{\underline{\rho }}]_{\tau })}dx}dt,\\ {\mathcal {L}}_{3,2}:=&\frac{1}{2\tau }\int _0^{t_m}{\int _{\varOmega }{(\rho _h-{\underline{\rho }}_{\tau }){\mathcal {H}}'''(\xi _{{\underline{\rho }}})({\underline{\rho }}_{\tau }-{\underline{\rho }}_{\tau }^{\star })^2}dx}dt. \end{aligned}$$Using the Taylor formula and the mass conservation (see, Remark 3.1), we obtain
$$\begin{aligned} |{\mathcal {L}}_{3,1}|\le&\tau C({\underline{\rho }}_{\min },{\underline{\rho }}_{\max })|p'|_{C^1([{\underline{\rho }}_{\min },{\underline{\rho }}_{\max }])}\Vert {\underline{\rho }}_{\tau }-\rho _h\Vert _{L^1((0,T)\times \varOmega )}\Vert \partial _{tt}{\underline{\rho }}\Vert _{L^\infty ((0,T)\times \varOmega )}\\ \le&\tau C(M_0,{\underline{\rho }}_{\min },{\underline{\rho }}_{\max },|p'|_{C^1([{\underline{\rho }}_{\min },{\underline{\rho }}_{\max }])},\Vert \partial _{tt}{\underline{\rho }}\Vert _{L^\infty ((0,T)\times \varOmega )}). \end{aligned}$$By a similar argument, we conclude that
$$\begin{aligned} |{\mathcal {L}}_{3,2}|\le \tau C(M_0,{\underline{\rho }}_{\min },{\underline{\rho }}_{\max },|p'|_{C^1([{\underline{\rho }}_{\min },{\underline{\rho }}_{\max }])},\Vert \partial _t{\underline{\rho }}\Vert _{L^\infty ((0,T)\times \varOmega )}). \end{aligned}$$ -
The term \({\mathcal {R}}_5\). We may write
$$\begin{aligned} {\mathcal {R}}_5=&-\int _0^{t_m}{\int _{\varOmega }{\rho _h\varvec{u}_h\cdot ({\mathcal {H}}''({\underline{\rho }}_{\tau }^{\star })\nabla {\underline{\rho }}_{\tau }^{\star }-{\mathcal {H}}''({\underline{\rho }}_{\tau })\nabla {\underline{\rho }}_{\tau })}dx}dt\\&-\int _0^{t_m}{\int _{\varOmega }{\rho _h\varvec{u}_h\cdot {\mathcal {H}}''({\underline{\rho }}_{\tau })\nabla {\underline{\rho }}_{\tau }}dx}dt={\mathcal {L}}_{3,3}+{\mathcal {L}}_2. \end{aligned}$$By applying the first-order Taylor formula, we obtain
$$\begin{aligned}&\Vert p'({\underline{\rho }}_{\tau }^\star )-p'({\underline{\rho }}_{\tau })\Vert _{L^\infty ((0,T)\times \varOmega )}\nonumber \\&\quad \le C(|p'|_{C^1([{\underline{\rho }}_{\min },{\underline{\rho }}_{\max }])},\Vert \partial _t{\underline{\rho }}\Vert _{L^\infty ((0,T)\times \varOmega )})\tau . \end{aligned}$$(7.14)Using the estimate (7.14) and the Taylor formula, we have
$$\begin{aligned}&\Vert {\mathcal {H}}''({\underline{\rho }}_{\tau }^{\star })-{\mathcal {H}}''({\underline{\rho }}_{\tau })\Vert _{L^\infty ((0,T)\times \varOmega )}\nonumber \\&\quad \le C({\underline{\rho }}_{\min },{\underline{\rho }}_{\max })\Vert p'({\underline{\rho }}_{\tau }^\star )-p'({\underline{\rho }}_{\tau })\Vert _{L^\infty ((0,T)\times \varOmega )}\nonumber \\&\qquad +C({\underline{\rho }}_{\min },{\underline{\rho }}_{\max },|p'|_{C^1([{\underline{\rho }}_{\min },{\underline{\rho }}_{\max }])})\Vert {\underline{\rho }}_{\tau }^\star -{\underline{\rho }}_{\tau }\Vert _{L^\infty ((0,T)\times \varOmega )}\nonumber \\&\quad \le C({\underline{\rho }}_{\min },{\underline{\rho }}_{\max },|p'|_{C^1([{\underline{\rho }}_{\min },{\underline{\rho }}_{\max }])},\Vert \partial _t{\underline{\rho }}\Vert _{L^\infty ((0,T)\times \varOmega )})\tau . \end{aligned}$$(7.15)Therefore, by using the Hölder inequality and the estimate (7.15), we obtain
$$\begin{aligned} |{\mathcal {L}}_{3,3}|\le&C(\varOmega ,T)\Vert \rho _h\Vert _{L^{2\gamma }((0,T)\times \varOmega )}\Vert \varvec{u}_h\Vert _{L^2(0,T;\varvec{H}^1(\varOmega ))}\\&\times \Vert ({\mathcal {H}}''({\underline{\rho }}_{\tau }^{\star })-{\mathcal {H}}''({\underline{\rho }}_{\tau }))\nabla {\underline{\rho }}_{\tau }^{\star }\Vert _{L^\infty ((0,T)\times \varOmega )}\\ \le&C(\varOmega ,T)\Vert \rho _h\Vert _{L^{2\gamma }((0,T)\times \varOmega )}\Vert \varvec{u}_h\Vert _{L^2(0,T;\varvec{H}^1(\varOmega ))}\\&\times \Vert {\mathcal {H}}''({\underline{\rho }}_{\tau })\nabla ({\underline{\rho }}_{\tau }^{\star }-{\underline{\rho }}_{\tau })\Vert _{L^\infty ((0,T)\times \varOmega )}\\ \le&C(\varOmega ,T,E_0,{\underline{\rho }}_{\min },{\underline{\rho }}_{\max },|p'|_{C^1([{\underline{\rho }}_{\min },{\underline{\rho }}_{\max }])},\Vert (\partial _t{\underline{\rho }},\nabla {\underline{\rho }},\partial _t\nabla {\underline{\rho }})\Vert _{L^\infty ((0,T)\times \varOmega )})\tau . \end{aligned}$$ -
The term \({\mathcal {R}}_6\). By applying the estimate (4.3) of Lemma 4.2, we get that
$$\begin{aligned} |{\mathcal {R}}_6|\le&C(\varOmega ,T)\Vert \nabla {\mathcal {H}}'({\underline{\rho }}_{\tau }^{\star })\Vert _{L^\infty ((0,T)\times \varOmega )}h^A\\ \le&C(\varOmega ,T,{\underline{\rho }}_{\min },{\underline{\rho }}_{\max },|p'|_{C^1([{\underline{\rho }}_{\min },{\underline{\rho }}_{\max }])},\Vert \nabla {\underline{\rho }}\Vert _{L^\infty ((0,T)\times \varOmega )})h^A. \end{aligned}$$
Combining the above analysis, we have \(\sum _{i=1}^6{\mathcal {R}}_i={\mathcal {R}}_1+{\mathcal {R}}_3+\sum _{i=1}^3{\mathcal {L}}_i\). After setting \({\mathcal {L}}_3:=\sum _{i=1}^3{\mathcal {L}}_{3,i}+{\mathcal {R}}_2+{\mathcal {R}}_6\), we deduce the approximate relative energy inequality (7.12) from the estimate (7.5). The proof is thus complete. \(\square \)
Next we derive a discrete identity for the strong solutions.
Theorem 7.5
Suppose that Hypothesis 2.1 is satisfied and the pressure \(p=p(\rho )\) satisfies the hypothesis (1.2) with \(\gamma >1\). Let the internal energy \({\mathcal {H}}\) is given by \({\mathcal {H}}(\rho )=\frac{p(\rho )}{\gamma -1}\). Let the family \(({\underline{\rho }}_{\tau },\underline{\varvec{u}}_{\tau })\) be defined as in (7.3). For any \(1\le m\le N\), then the following identity holds:
where the remainder terms \({\mathcal {L}}_i\) are defined by
Proof
Since \(({\underline{\rho }},\underline{\varvec{u}})\) is a strong solution of the problem (1.1), the second equation of (1.1) can be rewritten in the form
Taking \(t=t_n\) in (7.17), multiplying this identity by \(\varvec{u}_h^n-\underline{\varvec{u}}^n\) and integration over \(\varOmega \). We get, after summation from \(n=1\) to \(n=m\),
which implies that \({\mathcal {R}}_1+\sum _{i=4}^5{\mathcal {L}}_i=0\). The proof is thus complete. \(\square \)
Now, we will derive the unconditional error estimate of the problem (1.1) based on the approximate relative energy inequality (7.13) and the discrete identity (7.16).
Theorem 7.6
Let \(\varOmega \subset {\mathbb {R}}^d\), \(d=2,3\) be a bounded domain and assume that the viscosity coefficient \(\mu ,\lambda \) satisfies \(\mu >0\) and \(d\lambda +2\mu >0\). Suppose that the pressure \(p=p(\rho )\) satisfies the assumption (1.2) with \(\gamma >\frac{6}{5}\). The initial values \((\rho _0,\varvec{u}_0)\) satisfies Hypothesis 2.1 with the finite energy \(E_0:=\int _{\varOmega }{{\mathcal {H}}(\rho _0)}dx\) and finite mass \(M_0:=\int _{\varOmega }{\rho _0}dx\). Let \(({\underline{\rho }},\underline{\varvec{u}})\) be a strong solution of the problem (1.1) belonging to the class
emanating from the initial data \(({\underline{\rho }}_0,\underline{\varvec{u}}_0)\). Let the families \((\rho _h,\varvec{u}_h)\) and \(({\underline{\rho }}_{\tau },\underline{\varvec{u}}_{\tau })\) be defined as in (3.16)–(3.17) and (7.3), respectively. Then there exists
such that for any \(1\le m\le N\), then we have
Proof
Combining the approximate relative energy inequality (7.12) and the discrete identity (7.16), we can show
where the terms \({\mathcal {L}}_i\) are defined by
We next bound the term \({\mathcal {L}}_6\). Since the pair \(({\underline{\rho }},\underline{\varvec{u}})\) is a strong solution of the problem (1.1), the first equation of (1.1) can be rewritten in the form
By the identity (7.19), we write
where
It is easy to check that
Let \(\varOmega _{h,1}:=\{\frac{{\underline{\rho }}_{\tau }}{2}<\rho _h<2{\underline{\rho }}_{\tau }\}\) and \(\varOmega _{h,2}:=\varOmega \setminus \varOmega _{h,1}\). The term \({\mathcal {L}}_{6,2}\) can be rewritten as
where
By applying the Poincaré and Young inequalities, the estimate (7.2), \({\underline{\rho }}_{\tau }\in ({\underline{\rho }}_{\min },{\underline{\rho }}_{\max })\), we can show
By employing the estimate (7.2) and \({\underline{\rho }}_{\tau }\in ({\underline{\rho }}_{\min },{\underline{\rho }}_{\max })\), \(\gamma >\frac{6}{5}\), we have
Using the Poincaré and Young inequalities, the estimates (7.20) and (7.4), \(\underline{\rho }_\tau \in (\underline{\rho }_{\min },\underline{\rho }_{\max })\), we conclude that
Combining the above analysis with \(\delta =\frac{\mu }{4}\), we get that
where the constant \(C>0\) is given by
Using the estimate (7.4), we can show
which implies that
By applying the standard discrete version of Gronwall’s lemma for the inequality (7.21), the proof is thus complete. \(\square \)
Finally, we will give an error estimate for the discrete \(L^2(L^2)\) norm of \(p(\rho _h)\).
Theorem 7.7
Suppose that the condition of Theorem 7.6 holds. Let the families \((\rho _h,\varvec{u}_h)\) and \(({\underline{\rho }}_{\tau },\underline{\varvec{u}}_{\tau })\) be defined as in (3.16)–(3.17) and (7.3), respectively. Then there exists
such that for any \(1\le m\le N\), we have
Proof
Taking \(t=t_n\) in (7.17), multiplying this identity by \(\varvec{v}_h\in {\mathbb {V}}_h\) and integral over \(\varOmega \), we conclude that
Subtracting (7.23) from (3.8), we can get the error equation
Let \(r_{\rho }^n:=(p(\rho _h^n)-p({\underline{\rho }}_h^n))-\frac{1}{|\varOmega |}\int _{\varOmega }{(p(\rho _h^n)-p({\underline{\rho }}_h^n))}dx\) for \(1\le n\le N\). Taking \(\varvec{v}_h=\varPi _h^{{\mathbb {V}}}{\textbf{B}}[r_{\rho }^n]\) in (7.24), we can show
By applying the Cauchy–Schwarz inequality, the estimates (3.3) and (3.12), we obtain
which implies that
Summing (7.25) from \(n=1\) to \(n=m\) and multiplying the resulting inequality by \(\tau \), we conclude that
where the terms \({\mathcal {L}}_i\) (\(8\le i\le 10\)) are defined by
Bound on \({\mathcal {L}}_9\). By applying the estimate (3.1), the mean value theorem and \({\underline{\rho }}^n,{\underline{\rho }}_h^n\in [{\underline{\rho }}_{\min },{\underline{\rho }}_{\max }]\), we can show
Let \(\varOmega _{h,1}^n:=\{\frac{{\underline{\rho }}^n}{2}<\rho _h^n<2{\underline{\rho }}^n\}\) and \(\varOmega _{h,2}^n:=\varOmega \setminus \varOmega _{h,1,n}\). By applying the estimate (7.2) and (7.20), we obtain
Using the estimate (7.4), we get that
Combining the error estimate of Theorem 7.6, which implies that
By a similar argument, we can show
Combining the above analysis, the proof is thus complete. \(\square \)
References
Agmon, S.: Lectures on elliptic boundary value problems. Prepared for publication by B. Frank Jones, Jr. with the assistance of George W. Batten, Jr. Van Nostrand Mathematical Studies, No. 2. D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London (1965)
Akbas, M., Gallouët, T., Gassmann, A., Linke, A., Merdon, C.: A gradient-robust well-balanced scheme for the compressible isothermal Stokes problem. Comput. Methods Appl. Mech. Engrg. 367, 113069 (2020)
Bernardi, C., Raugel, G.: Analysis of some finite elements for the Stokes problem. Math. Comp. 44(169), 71–79 (1985)
Bogovskiĭ, M.E.: Solutions of some problems of vector analysis, associated with the operators \({\rm div}\) and \({\rm grad}\). In Theory of cubature formulas and the application of functional analysis to problems of mathematical physics, volume 1980 of Trudy Sem. S. L. Soboleva, No. 1, pp 5–40, 149. Akad. Nauk SSSR Sibirsk. Otdel., Inst. Mat., Novosibirsk (1980)
Brenner, S.C., Ridgway Scott, L.: The mathematical theory of finite element methods, volume 15 of Texts in Applied Mathematics. Springer, New York (1994)
DiPerna, R.J., Lions, P.-L.: Ordinary differential equations, transport theory and Sobolev spaces. Invent. Math. 98(3), 511–547 (1989)
Dreher, M., Jüngel, A.: Compact families of piecewise constant functions in \(L^p(0, T;B)\). Nonlinear Anal. 75(6), 3072–3077 (2012)
Evans, L.C.: Partial differential equations. Graduate Studies in Mathematics, vol. 19. American Mathematical Society, Providence, RI (1998)
Eymard, R., Gallouët, T., Herbin, R., Latché, J.-C.: Convergence of the MAC scheme for the compressible Stokes equations. SIAM J. Numer. Anal. 48(6), 2218–2246 (2010)
Eymard, R., Gallouët, T., Herbin, R., Latché, J.C.: A convergent finite element-finite volume scheme for the compressible Stokes problem. II. The isentropic case. Math. Comp. 79(270), 649–675 (2010)
Feireisl, E.: Dynamics of viscous compressible fluids. Oxford Lecture Series in Mathematics and its Applications, vol. 26. Oxford University Press, Oxford (2004)
Feireisl, E., Hošek, R., Maltese, D., Novotný, A.: Error estimates for a numerical method for the compressible Navier–Stokes system on sufficiently smooth domains. ESAIM, Math. Model. Numer. Anal. 51(1), 279–319 (2017)
Feireisl, E., Jin, B.J., Novotný, A.: Relative entropies, suitable weak solutions, and weak-strong uniqueness for the compressible Navier-Stokes system. J. Math. Fluid Mech. 14(4), 717–730 (2012)
Feireisl, E., Karper, T., Novotný, A.: A convergent numerical method for the Navier–Stokes–Fourier system. IMA J. Numer. Anal. 36(4), 1477–1535 (2016)
Feireisl, E., Karper, T.G., Pokorný, M.: Mathematical theory of compressible viscous fluids. Advances in Mathematical Fluid Mechanics. Birkhäuser/Springer, Cham (2016). Analysis and Numerics, Lecture Notes in Mathematical Fluid Mechanics
Feireisl, E., Novotný, A.: Singular limits in thermodynamics of viscous fluids. Advances in Mathematical Fluid Mechanics. Birkhäuser Verlag, Basel (2009)
Feireisl, E., Novotný, A., Sun, Y.: Suitable weak solutions to the Navier-Stokes equations of compressible viscous fluids. Indiana Univ. Math. J. 60(2), 611–631 (2011)
Fettah, A., Gallouët, T.: Numerical approximation of the general compressible Stokes problem. IMA J. Numer. Anal. 33(3), 922–951 (2013)
Galdi, G.P.: An introduction to the mathematical theory of the Navier-Stokes equations. Vol. I, volume 38 of Springer Tracts in Natural Philosophy. Springer, New York (1994). Linearized steady problems
Gallouët, T., Herbin, R., Latché, J.-C.: A convergent finite element-finite volume scheme for the compressible Stokes problem. I. The isothermal case. Math. Comp. 78(267), 1333–1352 (2009)
Gallouët, T., Herbin, R., Maltese, D., Novotny, A.: Convergence of the marker-and-cell scheme for the semi-stationary compressible Stokes problem. Math. Comput. Simul. 137, 325–349 (2017)
Gallouët, T., Gastaldo, L., Herbin, R., Latché, J.-C.: An unconditionally stable pressure correction scheme for the compressible barotropic Navier–Stokes equations. M2AN Math. Model. Numer. Anal. 42(2), 303–331 (2008)
Gallouët, T., Herbin, R., Maltese, D., Novotny, A.: Error estimates for a numerical approximation to the compressible barotropic Navier–Stokes equations. IMA J. Numer. Anal. 36(2), 543–592 (2016)
Gallouët, T., Maltese, D., Novotny, A.: Error estimates for the implicit MAC scheme for the compressible Navier–Stokes equations. Numer. Math. 141(2), 495–567 (2019)
Girault, V., Raviart, P.-A.: Finite element methods for Navier–Stokes equations, volume 5 of Springer Series in Computational Mathematics. Springer, Berlin (1986) Theory and algorithms
Karlsen, K.H., Karper, T.K.: A convergent nonconforming finite element method for compressible Stokes flow. SIAM J. Numer. Anal. 48(5), 1846–1876 (2010)
Karlsen, K.H., Karper, T.K.: Convergence of a mixed method for a semi-stationary compressible Stokes system. Math. Comp. 80(275), 1459–1498 (2011)
Karper, T.K.: A convergent FEM-DG method for the compressible Navier–Stokes equations. Numer. Math. 125(3), 441–510 (2013)
Kwon, Y.-S., Novotný, A.: Construction of weak solutions to compressible Navier–Stokes equations with general inflow/outflow boundary conditions via a numerical approximation. Numer. Math. 149(4), 717–778 (2021)
Lions, P.-L.: Mathematical topics in fluid mechanics. Vol. 2, volume 10 of Oxford Lecture Series in Mathematics and its Applications. The Clarendon Press, Oxford University Press, New York (1998) Compressible models, Oxford Science Publications
Mizerová, H., She, B.: Convergence and error estimates for a finite difference scheme for the multi-dimensional compressible Navier–Stokes system. J. Sci. Comput. 84(1), Paper No. 25, 39 (2020)
Acknowledgements
The research was supported by National Natural Science Foundation of China (Nos. 11871467, 12271514 and 12161141017).
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Appendix
Appendix
1.1 The Proof of Theorem 3.3
Our goal is to show the existence of numerical solutions for the scheme (3.7)–(3.8) by applying Schaeffer’s fixed point theorem. For this purpose, we define the mapping
in the following way.
-
Given \(\varvec{u}\in {\mathbb {V}}_h\), we will prove the unique solution \(\rho \in {\mathbb {Q}}_h\) of the linear system
$$\begin{aligned} \int _{\varOmega }{\frac{\rho -\rho _h^{n-1}}{\tau }\varphi _h}dx&-\sum _{F\in {\mathcal {F}}_{h,int}}{\int _{F}{{\text {Up}}[\rho ,\varvec{u}]-h^{\epsilon -1}\llbracket {\rho }\rrbracket \llbracket {\varphi _h}\rrbracket }dS}\nonumber \\&+h^{\epsilon -1}\sum _{F\in {\mathcal {F}}_{h,int}}{\int _{F}{\llbracket {\rho }\rrbracket \llbracket {\varphi _h}\rrbracket }dS}=0, \end{aligned}$$(A.1)for any \(\varphi _h\in {\mathbb {Q}}_h\). In order to prove the linear problem (A.1) has a unique solution \(\rho (\varvec{u})\), we need prove that the associated homogenous problem
$$\begin{aligned} \int _{\varOmega }{\rho \varphi _h}dx-\tau \sum _{F\in {\mathcal {F}}_{h,int}}{\int _{F}{{\text {Up}}\left[ \rho ,\varvec{u}\right] \llbracket {\varphi _h}\rrbracket }dS}+h^{\epsilon -1}\tau \sum _{F\in {\mathcal {F}}_{h,int}}{\int _{F}{\llbracket {\rho }\rrbracket \llbracket {\varphi _h}\rrbracket }dS}=0 \end{aligned}$$(A.2)admits a unique solution \(\rho =0\). By the same proof of [14, Section 4.3], we can show the homogenous problem (A.2) of renormalized equation
$$\begin{aligned}&\int _{\varOmega }{{\mathcal {B}}'(\rho )\rho \varphi _h}dx-\tau \sum _{F\in {\mathcal {F}}_{h,int}}{\int _{F}{{\text {Up}}[{\mathcal {B}}(\rho ),\varvec{u}]\llbracket {\varphi _h}\rrbracket }dS}\nonumber \\&+h^{\epsilon -1}\tau \sum _{F\in {\mathcal {F}}_{h,int}}{\int _{F}{{\mathcal {B}}'(\rho _{+})\llbracket \rho \rrbracket \llbracket \varphi _h\rrbracket }dS}+h^{\epsilon -1}\tau \sum _{F\in {\mathcal {F}}_{h,int}}{\int _{F}{{\mathcal {B}}''({\overline{\eta }}_{\rho })\llbracket \rho \rrbracket ^2}dS}\nonumber \\&+\frac{\tau }{2}\sum _{F\in {\mathcal {F}}_{h,int}}{\int _{F}{\varphi _{h}{\mathcal {B}}''(\eta _{\rho })\llbracket \rho \rrbracket ^2|\varvec{u}\cdot \varvec{n}|}dS}=\tau \int _{\varOmega }{\varphi _h({\mathcal {B}}(\rho )-{\mathcal {B}}'(\rho )\rho ){\text {div}}\varvec{u}}dx, \end{aligned}$$(A.3)for any \(\varphi _h\in {\mathbb {Q}}_h\), where \({\mathcal {B}}\in C^2({\mathbb {R}}_{+})\), \({\overline{\eta }}_{\rho },\eta _{\rho }\in {\text {co}}\{\rho ,\rho _{+}\}\) on each face \(F\in {\mathcal {F}}_h\). Any non negative \(C^2({\mathbb {R}})\) convex approximations function \({\mathcal {S}}_{\epsilon }\) such that \({\mathcal {S}}_{\epsilon }(\rho )\rightarrow {\mathcal {S}}(\rho )\) and \({\mathcal {S}}_{\epsilon }'(\rho )\rightarrow {\mathcal {S}}'(\rho )\) for all \(\rho \ne 0\), where \({\mathcal {S}}(\rho )=\max \{-\rho ,0\}\). Taking \((\varphi _h,{\mathcal {B}})=(1,{\mathcal {S}}_{\epsilon })\) in (A.3), we have
$$\begin{aligned} \int _{\varOmega }{{\mathcal {S}}_{\epsilon }(\rho )}dx\le \tau \int _{\varOmega }{\varphi _h({\mathcal {S}}_{\epsilon }(\rho )-{\mathcal {S}}_{\epsilon }'(\rho )\rho ){\text {div}}\varvec{u}}dx+\int _{\varOmega }{({\mathcal {S}}_{\epsilon }(\rho )-{\mathcal {S}}_{\epsilon }'(\rho )\rho )}dx. \end{aligned}$$(A.4)Combining the inequality (A.4) and \({\mathcal {S}}(\rho )-{\mathcal {S}}'(\rho )\rho =0\) for all \(\rho \ne 0\), we obtain \({\mathcal {S}}(\rho )=0\) and \(\rho \ge 0\). Let \(\varphi _h=1\) in (A.2), we obtain
$$\begin{aligned} \int _{\varOmega }{\rho }dx=0. \end{aligned}$$(A.5)According to \(\rho \ge 0\) and (A.5), we have \(\rho =0\), then the problem (A.1) has a unique solution \(\rho (\varvec{u})\). By applying the Lemma 3.3, we have \(\rho (\varvec{u})>0\).
-
For given \(\rho \in {\mathbb {Q}}_h\) and \(\varvec{u}\in {\mathbb {V}}_h\), we can show the unique solution \(\varvec{U}\in {\mathbb {V}}_h\) of the algebraic system
$$\begin{aligned} \int _{\varOmega }{[\mu \nabla \varvec{U}:\nabla \varvec{v}_h+(\lambda +\mu ){\text {div}}\varvec{U}{\text {div}}\varvec{v}_h]}dx=\int _{\varOmega }{p(\rho ){\text {div}}\varvec{v}_h}dx, \end{aligned}$$(A.6)for any \(\varvec{v}_h\in {\mathbb {V}}_h\), where \(\rho =\rho (\varvec{u})\) is determined by (A.1). Similarly, by applying the Lax-Milgram Lemma for the linear system (A.6), we have a unique solution \(\varvec{U}\in {\mathbb {V}}_h\).
Clearly, any fixed point of the mapping \({\mathcal {L}}\) is a solution of the scheme (3.7)–(3.8). Next, we need show that the set
satisfies the conditions of Lemma 3.5. In other words, we need to verify that the set is non empty and bounded. It is obvious show that the set is non empty (\({\textbf{0}}\) belongs to the set). Finally, for all \(\varLambda \in (0,1]\), we need to prove the solution \(\varvec{u}\) of the equation \(\varvec{u}=\varLambda {\mathcal {L}}[\varvec{u}]\) can be bounded in terms of the local data \((\rho _h^{n-1},\varvec{u}_h^{n-1})\) uniformly with respect to \(\varLambda \). Setting \(\rho _h^n=\rho (\varvec{u})\), \(\varvec{u}_h^n=\varvec{u}\) in (3.7)–(3.8), where \(\varvec{u}\) is a solution of \(\varvec{u}=\varLambda {\mathcal {L}}[\varvec{u}]\), we have
By recalling the steps in the proof of discrete energy estimate (3.10), we can show
Combining (A.7) and \(0<\varLambda \le 1\), there exists a constant C independent of \(\varLambda \) such that
Combining the above conclusions and Lemma 3.5, we can show the schemes (3.7)–(3.8) has at least one solution. By applying the Lemma 3.3, we obtain the density \(\rho _h^n>0\). The proof is thus complete.
1.2 The Proof of Theorem 6.1
Taking the zero extension of \(\varvec{v}_h\) for \({\mathbb {R}}^d\setminus \varOmega \). We show the proof of this Theorem in two steps. Step 1. If \(q=2\), for any \(\varvec{x}\in {\mathbb {R}}^d\), it is easy to check that
For the identity (A.8), by applying Cauchy–Schwarz inequality, we conclude that
Therefor, by employing Fubini theorem and \(\nabla \varvec{v}_h\) vanishes outside \(\varOmega \), we have
Step 2. For the case of \(2<q\le 6\), by applying Gagliardo-Nirenberg interpolation inequality and (A.9), we obtain
According to the embedding \(\varvec{H}_0^1\hookrightarrow \varvec{L}^6\) and the Poincaré inequality, we get
Inserting (A.11) into (A.10), which implies that
Combining the inequalities (A.9) and (A.12), the proof is thus complete.
1.3 Some Functional Analysis Results
For the convenience of readers, we list some functional analysis results that need to be used in this article. We first recall the following weak convergence and monotonicity properties (see, e.g., [16, Theorem 10.19]):
Lemma A.1
Let \(I\subset {\mathbb {R}}\) be an interval, \(Q\subset {\mathbb {R}}^N\) a domain, and \((P,G)\in C(I)\times C(I)\) a couple of non-decreasing functions. Assume that \(\rho _n\in L^1(Q;I)\) is a sequence such that
(i) Then \(\overline{P(\rho )}\;\overline{G(\rho )}\le \overline{P(\rho )G(\rho )}\). (ii) If, in addition, \(G\in C({\mathbb {R}})\), \(G({\mathbb {R}})={\mathbb {R}}\), G is strictly increasing, \(P\in C({\mathbb {R}})\), P is non-decreasing, and \(\overline{P(\rho )}\;\overline{G(\rho )}=\overline{P(\rho )G(\rho )}\), then \(\overline{P(\rho )}=P\circ G^{-1}(\overline{G(\rho )})\). (iii) In particular, if \(G(z)=z\), then \(\overline{P(\rho )}=P(\rho )\).
Secondly, the convex function have the lower semi-continuous with respect to the weak topology on \(L^1(O)\) (see, e.g., [11, Theorem 2.11]).
Lemma A.2
Let \(O\subset {\mathbb {R}}^N\) be a measurable set and \(\{\varvec{v}_n\}_{n=1}^\infty \) a sequence of functions in \(L^1(O;{\mathbb {R}}^M)\) such that
Let \(\varPhi :{\mathbb {R}}^M\rightarrow (-\infty ,\infty ]\) be a lower semi-continuous convex function such that \(\varPhi (\varvec{v}_n)\in L^1(O)\) for any n, and
Then
If, moreover, \(\varPhi \) is strictly on an open convex set \(U\subset {\mathbb {R}}^M\), and
then
extracting subsequence as the case may be.
Next, we introduce the following sequential compactness (see, e.g., [15, Lemma 3]).
Lemma A.3
Let \(Q\subset {\mathbb {R}}^M\), suppose that \(\rho _n\rightharpoonup \rho \) in \(L^2(Q)\) and \(\overline{\rho \log (\rho )}=\rho \log (\rho )\) are satisfied. Then
Finally, we recall the following discrete version of the Aubin-Lions compactness Lemma for the Bochner spaces, which is useful in the convergence analysis. (see, e.g., [7, Theorem 1]).
Lemma A.4
Let \({\mathbb {E}}_0\), \({\mathbb {E}}\) and \({\mathbb {E}}_1\) be Banach spaces such that the embedding \({\mathbb {E}}_0\hookrightarrow {\mathbb {E}}\) is compact and \({\mathbb {E}}\hookrightarrow {\mathbb {E}}_1\) is continuous. Given \(T>0\) and a small number \(\tau >0\), write \((0,T]=\cup _{k=1}^M(t_{k-1},t_k]\) with \(t_k=k\tau \) and \(M\tau =T\). Let \(\{v_\tau \}_{\tau >0}\) be a sequence such that
-
The mapping \(t\mapsto v_\tau (t,\cdot )\) is constant on each interval \((t_{k-1},t_k]\), \(k=1,2,\ldots ,M\).
-
Let \(D_tv_\tau (t,\cdot )=(v_\tau (t,\cdot )-v_\tau (t-\tau ,\cdot ))/\tau \) be the discrete time derivative of \(v_\tau (t,\cdot )\). The sequence \(\{v_\tau \}_{\tau >0}\) satisfies the following estimates:
$$\begin{aligned} \Vert v_\tau \Vert _{L^{p_0}(0,T;{\mathbb {E}}_0)}+\Vert D_tv_\tau \Vert _{L^{p_1}(\tau ,T;{\mathbb {E}}_1)}\le C, \end{aligned}$$for any \(1<p_0,p_1<\infty \), where \(C_0\) is a constant which is independent of \(\tau \).
Then \(\{v_\tau \}_{\tau >0}\) is relatively compact in \(L^{p_0}(0,T;{\mathbb {E}})\).
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Mao, S., Xue, W. Convergence and Error Estimates of a Mixed Discontinuous Galerkin-Finite Element Method for the Semi-stationary Compressible Stokes System. J Sci Comput 94, 47 (2023). https://doi.org/10.1007/s10915-023-02096-7
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s10915-023-02096-7