Convergence and Error Estimates of a Mixed Discontinuous Galerkin-Finite Element Method for the Semi-stationary Compressible Stokes System

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.

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

$$\begin{aligned} {\mathcal {L}}:{\mathbb {V}}_h\rightarrow {\mathbb {V}}_h,\quad {\mathcal {L}}[\varvec{u}]\mapsto \varvec{U}, \end{aligned}$$

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

$$\begin{aligned} \{\varvec{z}\in {\mathbb {V}}_h:\varvec{z}=\varLambda {\mathcal {L}}(\varvec{z}),\;\varLambda \in [0,1]\} \end{aligned}$$

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

$$\begin{aligned} \int _{\varOmega }{d_t\rho _h^n\varphi _h}dx-\sum _{F\in {\mathcal {F}}_{h,int}}{\int _{F}{{\text {Up}}\left[ \rho _h^n,\varvec{u}_h^n\right] \llbracket {\varphi _h}\rrbracket }dS}\\ +h^{\epsilon -1}\sum _{F\in {\mathcal {F}}_{h,int}}{\int _{F}{\llbracket {\rho _h^n}\rrbracket \llbracket {\varphi _h}\rrbracket }dS}=0,\\ \varLambda ^{-1}\int _{\varOmega }{\left[ \mu \nabla \varvec{u}_h^n:\nabla \varvec{v}_h+(\lambda +\mu ){\text {div}}\varvec{u}_h^n{\text {div}}\varvec{v}_h\right] }dx-\int _{\varOmega }{p(\rho _h^n){\text {div}}\varvec{v}_h}dx=0. \end{aligned}$$

By recalling the steps in the proof of discrete energy estimate (3.10), we can show

$$\begin{aligned} \int _{\varOmega }{{\mathcal {H}}(\rho _h^n)}dx+\frac{1}{\varLambda }\int _{\varOmega }{[\mu |\nabla \varvec{u}_h^n|^2+(\lambda +\mu )|{\text {div}}\varvec{u}_h^n|^2]}dx\le \int _{\varOmega }{{\mathcal {H}}(\rho _h^{n-1})}dx. \end{aligned}$$

(A.7)

Combining (A.7) and $0<\varLambda \le 1$, there exists a constant C independent of $\varLambda$ such that

$$\begin{aligned} \Vert \varvec{u}_h^n\Vert _{{\mathbb {V}}_h}^2:=\mu \Vert \nabla \varvec{u}_h^n\Vert _{\varvec{L}^2(\varOmega )}^2\le C. \end{aligned}$$

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

$$\begin{aligned} \varvec{v}_h(\varvec{x})-\varvec{v}_h(\varvec{x}-\varvec{\xi })=\int _0^1{\nabla \varvec{v}_h(\varvec{x}-s\varvec{\xi })\cdot \varvec{\xi }}ds. \end{aligned}$$

(A.8)

For the identity (A.8), by applying Cauchy–Schwarz inequality, we conclude that

$$\begin{aligned} |\varvec{v}_h(\varvec{x})-\varvec{v}_h(\varvec{x}-\varvec{\xi })|^2\le |\varvec{\xi }|^2\int _0^1{|\nabla \varvec{v}_h(\varvec{x}-s\varvec{\xi })|^2}ds. \end{aligned}$$

Therefor, by employing Fubini theorem and $\nabla \varvec{v}_h$ vanishes outside $\varOmega$, we have

$$\begin{aligned} \int _{{\mathbb {R}}^d}{|\varvec{v}_h(\varvec{x})-\varvec{v}_h(\varvec{x}-\varvec{\xi })|^2}dx\le |\varvec{\xi }|^2\int _{\varOmega }{|\nabla \varvec{v}_h(\varvec{x})|^2}dx. \end{aligned}$$

(A.9)

Step 2. For the case of $2<q\le 6$, by applying Gagliardo-Nirenberg interpolation inequality and (A.9), we obtain

$$\begin{aligned} \Vert \varvec{v}_h(\cdot )-\varvec{v}_h(\cdot -\varvec{\xi })\Vert _{\varvec{L}^q({\mathbb {R}}^d)}\le&\Vert \varvec{v}_h(\cdot )-\varvec{v}_h(\cdot -\varvec{\xi })\Vert _{\varvec{L}^2({\mathbb {R}}^d)}^{\theta }\Vert \varvec{v}_h(\cdot )-\varvec{v}_h(\cdot -\varvec{\xi })\Vert _{\varvec{L}^6({\mathbb {R}}^d)}^{1-\theta }\nonumber \\ \le&|\varvec{\xi }|^{\theta }\Vert \nabla \varvec{v}_h\Vert _{\varvec{L}^2(\varOmega )}^{\theta }\Vert \varvec{v}_h(\cdot )-\varvec{v}_h(\cdot -\varvec{\xi })\Vert _{\varvec{L}^6({\mathbb {R}}^d)}^{1-\theta }. \end{aligned}$$

(A.10)

According to the embedding $\varvec{H}_0^1\hookrightarrow \varvec{L}^6$ and the Poincaré inequality, we get

$$\begin{aligned} \Vert \varvec{v}_h(\cdot )-\varvec{v}_h(\cdot -\varvec{\xi })\Vert _{\varvec{L}^6({\mathbb {R}}^d)}\le C\Vert \nabla \varvec{v}_h\Vert _{\varvec{L}^2(\varOmega )}. \end{aligned}$$

(A.11)

Inserting (A.11) into (A.10), which implies that

$$\begin{aligned} \Vert \varvec{v}_h(\cdot )-\varvec{v}_h(\cdot -\varvec{\xi })\Vert _{\varvec{L}^q({\mathbb {R}}^d)}\le C|\varvec{\xi }|^{\theta }\Vert \nabla \varvec{v}_h\Vert _{\varvec{L}^2(\varOmega )}. \end{aligned}$$

(A.12)

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

$$\begin{aligned} \left\{ \begin{aligned} P(\rho _n)&\rightharpoonup \overline{P(\rho )},\\ G(\rho _n)&\rightharpoonup \overline{G(\rho )},\\ P(\rho _n)G(\rho _n)&\rightharpoonup \overline{P(\rho )G(\rho )}, \end{aligned} \right\} \quad {\text {in}}\; L^1(Q). \end{aligned}$$

(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

$$\begin{aligned} \varvec{v}_n\rightharpoonup \varvec{v},\;{\text {in}}\; L^1(O;{\mathbb {R}}^M). \end{aligned}$$

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

$$\begin{aligned} \varPhi (\varvec{v}_n)\rightharpoonup \overline{\varPhi (\varvec{v})},\;{\text {in}}\; L^1(O). \end{aligned}$$

Then

$$\begin{aligned} \varPhi (\varvec{v})\le \overline{\varPhi (\varvec{v})}\;{\text {a.a.}}\;{\text {on}}\; O. \end{aligned}$$

If, moreover, $\varPhi$ is strictly on an open convex set $U\subset {\mathbb {R}}^M$, and

$$\begin{aligned} \varPhi (\varvec{v})=\overline{\varPhi (\varvec{v})}\;{\text {a.a.}}\;{\text {on}}\; O, \end{aligned}$$

then

$$\begin{aligned} \varvec{v}_n(\varvec{y})\rightarrow \varvec{v}(\varvec{y})\;{\text {for}}\;{\text {a.a.}}\;\varvec{y}\in \{\varvec{y}\in O:\varvec{v}(\varvec{y})\in U\} \end{aligned}$$

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

$$\begin{aligned} \rho _n\rightarrow \rho \;{\text {in}}\; L^1(Q). \end{aligned}$$

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 $L^{p_0}(0,T;{\mathbb {E}})$.