Analysis of a projection method for the Stokes problem using an $\varepsilon$-Stokes approach

Abstract

We generalize pressure boundary conditions of an $\varepsilon$-Stokes problem. Our $\varepsilon$-Stokes problem connects the classical Stokes problem and the corresponding pressure-Poisson equation using one parameter $\varepsilon >0$. For the Dirichlet boundary condition, it is proven in Matsui and Muntean (Adv Math Sci Appl, 27:181–191, 2018) that the solution for the $\varepsilon$-Stokes problem converges to the one for the Stokes problem as $\varepsilon$ tends to 0, and to the one for the pressure-Poisson problem as $\varepsilon$ tends to $\infty$. Here, we extend these results to the Neumann and mixed boundary conditions. We also establish error estimates in suitable norms between the solutions to the $\varepsilon$-Stokes problem, the pressure-Poisson problem and the Stokes problem, respectively. Several numerical examples are provided to show that several such error estimates are optimal in $\varepsilon$. Our error estimates are improved if one uses the Neumann boundary conditions. In addition, we show that the solution to the $\varepsilon$-Stokes problem has a nice asymptotic structure.

Appendix

Theorems 2.7, 4.2, 4.3, and Proposition 2.8 are generalizations of severaltheorems stated in [17]. In this appendix, however, we give their proofs for the readers’convenience. We define a continuous coercive bilinear form depending on $\varepsilon$ and prove Theorem 2.7 by the Lax–Milgram Theorem.

Proof of Theorem 2.7

We take arbitrary $u_1\in {H^1({\varOmega })}^n$ with $\gamma _0 u_1=u_b$. Since ${{\text {div}}}:{H^1_0({\varOmega })}^n\rightarrow {L^2({\varOmega })}/{{\mathbb {R}}}$ is surjective [9, Corollary 2.4, 2${}^\circ$], there exists $u_2\in {H^1_0({\varOmega })}^n$ such that ${{\text {div}}}u_2={{\text {div}}}u_1$. We put

$$\begin{aligned} u_0:=u_1-u_2, \end{aligned}$$

(A.1)

and note that $\gamma _0 u_0=u_b$ and ${{\text {div}}}u_0=0$. To simplify the notation, we set $u:=u_\varepsilon -u_0\in {H^1_0({\varOmega })}^n,p:=p_\varepsilon -p_0\in Q$, and define $f\in {H^{-1}({\varOmega })}^n$ and $g\in Q^*$ by

$$\begin{aligned} \begin{array}{rll} \langle f,v\rangle &{}{\displaystyle :=\int _{\varOmega }Fv-\int _{\varOmega }\nabla u_0:\nabla v-\int _{\varOmega }(\nabla p_0)\cdot v} &{}\text{ for } \text{ all } \;v\in {H^1_0({\varOmega })}^n,\\ \langle g,q\rangle &{}{\displaystyle :=\langle G,q\rangle -\int _{\varOmega }\nabla p_0\cdot \nabla q} &{}\text{ for } \text{ all } \;q\in Q. \end{array} \end{aligned}$$

(A.2)

Then, $(u_\varepsilon ,p_\varepsilon )$ satisfies (ES’) if and only if (u, p) satisfies

$$\begin{aligned} \left\{ \begin{array}{ll} {\displaystyle \int _{\varOmega }\nabla u:\nabla \varphi +\int _{\varOmega }(\nabla p)\cdot \varphi =\langle f,\varphi \rangle } &{} {\text{ for } \text{ all } } \;\varphi \in {H^1_0({\varOmega })}^n,\\ {\displaystyle \varepsilon \int _{\varOmega }\nabla p\cdot \nabla \psi +\int _{\varOmega }({{\text {div}}}u)\psi =\varepsilon \langle g,\psi \rangle } &{} {\text{ for } \text{ all } } \;\psi \in Q. \end{array}\right. \end{aligned}$$

(A.3)

Adding the equations in (A.3), we get

$$\begin{aligned} \begin{aligned} \left( \left( \begin{array}{c} u\\ p\end{array}\right) , \left( \begin{array}{c} \varphi \\ \psi \end{array}\right) \right) _\varepsilon&:=\int _{\varOmega }\nabla u:\nabla \varphi + \varepsilon \int _{\varOmega }\nabla p\cdot \nabla \psi +\int _{\varOmega }(\nabla p)\cdot \varphi + \int _{\varOmega }({{\text {div}}}u)\psi \\&= \langle f,\varphi \rangle +\varepsilon \langle g,\psi \rangle . \end{aligned} \end{aligned}$$

We check that $(\cdot ,\cdot )_\varepsilon$ is a continuous coercive bilinear form on ${H^1_0({\varOmega })}^n\times Q$. The bilinearity and continuity of $(\cdot ,\cdot )_\varepsilon$ are obvious. The coercivity of $(\cdot ,\cdot )_\varepsilon$ is obtained in the following way. Take $(v,q)^T\in {H^1_0({\varOmega })}^n\times Q$. We have the following sequence of inequalities:

$$\begin{aligned} \begin{aligned} \left( \left( \begin{array}{c} v\\ q\end{array}\right) , \left( \begin{array}{c} v\\ q\end{array}\right) \right) _\varepsilon&={\displaystyle \int _{\varOmega }\nabla v:\nabla v + \varepsilon \int _{\varOmega }\nabla q\cdot \nabla q +\int _{\varOmega }v\cdot \nabla q+\int _{\varOmega }({{\text {div}}}v)q}\\&= \Vert \nabla v\Vert ^2_{{L^2({\varOmega })}}+\varepsilon \Vert \nabla q\Vert ^2_{{L^2({\varOmega })}}\\&\ge {\displaystyle \mathrm{min}\{ 1,\varepsilon \}\left( \Vert \nabla v\Vert ^2_{{L^2({\varOmega })}}+\Vert \nabla q\Vert ^2_{{L^2({\varOmega })}}\right) }\\&\ge {\displaystyle c~\mathrm{min}\{ 1,\varepsilon \}\left( \Vert v\Vert ^2_{{H^1({\varOmega })}^n}+\Vert q\Vert ^2_{{H^1({\varOmega })}}\right) }. \end{aligned} \end{aligned}$$

Summarizing, $(\cdot ,\cdot )_\varepsilon$ is a continuous coercive bilinear form and ${H^1_0({\varOmega })}^n\times Q$ is a Hilbert space. Therefore, the conclusion of Theorem 2.7 follows from the Lax–MilgramTheorem. $\square$

Let $(u_S,p_S),(u_PP,p_PP)$ and $(u_\varepsilon ,p_\varepsilon )$ be the solutions of (S’), (PP’) and (ES’), respectively, as guaranteed by Theorems 2.5, 2.6 and 2.7. We show that the subtract $p_S-p_PP$ satisfies

$$\begin{aligned} {\varDelta }(p_S-p_PP)=0 \end{aligned}$$

in distributions sense. The weak harmonicity is the key ingredient to provingProposition 2.8.

Proof of Proposition 2.8

First, we prove that there exists a constant $c>0$ independent of $\varepsilon$ such that $\Vert u_S-u_PP\Vert _{{H^1({\varOmega })}^n} \le c\Vert \gamma _0 p_S-\gamma _0 p_PP\Vert _{H^{1/2}({\varGamma })},$ and if $\gamma _0(p_S-p_PP)=0$, then $p_PP=p_S$. Taking the divergence of the first equation of (S’), we obtain

$$\begin{aligned} {{\text {div}}}F={{\text {div}}}(-{\varDelta }u_S+\nabla p_S)=-{\varDelta }({{\text {div}}}u_S)+{\varDelta }p_S={\varDelta }p_S. \end{aligned}$$

in distributions sense. Since $p_S\in {H^1({\varOmega })}$ and $C^\infty _0({\varOmega })$ is dense in ${H^1_0({\varOmega })}$, it follows that

$$\begin{aligned} \int _{\varOmega }\nabla p_S\cdot \nabla \psi =-\int _{\varOmega }({{\text {div}}}F)\psi \end{aligned}$$

for all $\psi \in {H^1_0({\varOmega })}$. Together with (S’), (PP’) and ${H^1_0({\varOmega })}\subset Q$, we obtain

$$\begin{aligned} \left\{ \begin{array}{ll}{\displaystyle \int _{\varOmega }\nabla (u_S-u_PP):\nabla \varphi =-\int _{\varOmega }(\nabla (p_S-p_PP))\cdot \varphi } &{} {\text{ for } \text{ all } } \;\varphi \in {H^1_0({\varOmega })}^n, \\ {\displaystyle \int _{\varOmega }\nabla (p_S-p_PP)\cdot \nabla \psi =0} &{}{\text{ for } \text{ all } } \psi \in {H^1_0({\varOmega })}\end{array}\right. \end{aligned}$$

(A.4)

from the assumption $\langle G,\psi \rangle =\int _{\varOmega }\nabla F\cdot \psi$. Putting $\varphi :=u_S-u_PP\in {H^1_0({\varOmega })}^n$ in (A.4), we get

$$\begin{aligned} \begin{aligned} \Vert \nabla (u_S-u_PP)\Vert ^2_{{L^2({\varOmega })}^{n\times n}}&={\displaystyle -\int _{\varOmega }(\nabla (p_S-p_PP))\cdot (u_S-u_PP)}\\&\le \Vert \nabla (p_S-p_PP)\Vert _{{L^2({\varOmega })}^n}\Vert u_S-u_PP\Vert _{{L^2({\varOmega })}^n}. \end{aligned} \end{aligned}$$

Hence,

$$\begin{aligned} \Vert u_S-u_PP\Vert _{{H^1({\varOmega })}^n}\le c_1\Vert \nabla (p_S-p_PP)\Vert _{{L^2({\varOmega })}^n} \end{aligned}$$

(A.5)

holds. From the second equation of (A.4), we obtain

$$\begin{aligned}&\Vert \nabla (p_S-p_PP-\psi )\Vert ^2_{{L^2({\varOmega })}^n}\\&\quad ={\displaystyle \Vert \nabla (p_S-p_PP)\Vert ^2_{{L^2({\varOmega })}^n}+\Vert \nabla \psi \Vert ^2_{{L^2({\varOmega })}^n} -2\int _{\varOmega }\nabla (p_S-p_PP)\cdot \nabla \psi }\\&\quad =\Vert \nabla (p_S-p_PP)\Vert ^2_{{L^2({\varOmega })}^n}+\Vert \nabla \psi \Vert ^2_{{L^2({\varOmega })}^n}\\&\quad \ge \Vert \nabla (p_S-p_PP)\Vert ^2_{{L^2({\varOmega })}^n} \end{aligned}$$

for all $\psi \in {H^1_0({\varOmega })}$. Thus we find

$$\begin{aligned} \Vert \nabla (p_S-p_PP)\Vert _{{L^2({\varOmega })}^n} \le \inf _{\psi \in {H^1_0({\varOmega })}^n}\left( \Vert \nabla (p_S-p_PP-\psi )\Vert _{{L^2({\varOmega })}^n}\right) . \end{aligned}$$

Since $\gamma _0$ is surjective and the space $\text{ Ker }(\gamma _0)={H^1_0({\varOmega })}$, ${H^1({\varOmega })}/{H^1_0({\varOmega })}$ and $H^{1/2}({\varGamma })$ are isomorphic, there exists a constant $c_2>0$ such that $\Vert q\Vert _{{H^1({\varOmega })}/{H^1_0({\varOmega })}}\le c_2\Vert \gamma _0 q\Vert _{H^{1/2}({\varGamma })}$ for all $q\in {H^1({\varOmega })}$. Hence, we obtain

$$\begin{aligned} \begin{aligned} \Vert \nabla (p_S-p_PP)\Vert _{{L^2({\varOmega })}^n}&\le {\displaystyle \inf _{\psi \in {H^1_0({\varOmega })}^n}\Vert \nabla (p_S-p_PP-\psi )\Vert _{{L^2({\varOmega })}^n}}\\&\le {\displaystyle \inf _{\psi \in {H^1_0({\varOmega })}^n}\Vert p_S-p_PP-\psi \Vert _{H^1({\varOmega })}}\\&=\Vert p_S-p_PP\Vert _{{H^1({\varOmega })}/{H^1_0({\varOmega })}}\\&\le c_2\Vert \gamma _0 p_S-\gamma _0 p_PP\Vert _{H^{1/2}({\varGamma })}. \end{aligned} \end{aligned}$$

(A.6)

Together with (A.5), we obtain $\Vert u_S-u_PP\Vert _{{H^1({\varOmega })}^n} \le c_1c_2\Vert \gamma _0 p_S-\gamma _0 p_PP\Vert _{H^{1/2}({\varGamma })}.$ Moreover, if $\gamma _0(p_S-p_PP)=0$, then $p_PP=p_S$.

Next, we prove that there exists a constant $c>0$ independent of $\varepsilon$ such that $\Vert u_S-u_\varepsilon \Vert _{{H^1({\varOmega })}^n} \le c\Vert \gamma _0 p_S-\gamma _0 p_\varepsilon \Vert _{H^{1/2}({\varGamma })},$ and if $\gamma _0(p_S-p_PP)=0$, then $p_PP=p_\varepsilon$. Let $w_\varepsilon :=u_S-u_\varepsilon \in {H^1_0({\varOmega })}^n$ and $r_\varepsilon :=p_PP-p_\varepsilon \in Q$. By (S’), (PP’) and (ES’), we obtain

$$\begin{aligned} \left\{ \begin{array}{ll} {\displaystyle \int _{\varOmega }\nabla w_\varepsilon :\nabla \varphi +\int _{\varOmega }(\nabla r_\varepsilon )\cdot \varphi =-\int _{\varOmega }(\nabla (p_S-p_PP))\cdot \varphi }&{} {\text{ for } \text{ all } } \;\varphi \in {H^1_0({\varOmega })}^n, \\ {\displaystyle \varepsilon \int _{\varOmega }\nabla r_\varepsilon \cdot \nabla \psi +\int _{\varOmega }({{\text {div}}}w_\varepsilon )\psi =0} &{}{\text{ for } \text{ all } } \;\psi \in Q. \end{array}\right. \end{aligned}$$

(A.7)

Putting $\varphi :=w_\varepsilon$ and $\psi :=r_\varepsilon$ and adding the two equations of (A.7), we get

$$\begin{aligned} \Vert \nabla w_\varepsilon \Vert ^2_{{L^2({\varOmega })}^{n\times n}}+\varepsilon \Vert \nabla r_\varepsilon \Vert ^2_{{L^2({\varOmega })}^n} \le \Vert \nabla (p_S-p_PP)\Vert _{{L^2({\varOmega })}^n}\Vert w_\varepsilon \Vert _{{L^2({\varOmega })}^n} \end{aligned}$$

(A.8)

from $\int _{\varOmega }(\nabla r_\varepsilon )\cdot w_\varepsilon =-\int _{\varOmega }({{\text {div}}}w_\varepsilon )r_\varepsilon$. Thus we find

$$\begin{aligned} \Vert w_\varepsilon \Vert _{{H^1({\varOmega })}^n}\le c_3\Vert \nabla (p_S-p_PP)\Vert _{{L^2({\varOmega })}^n}. \end{aligned}$$

Together with (A.6), we obtain

$$\begin{aligned} \Vert u_S-u_\varepsilon \Vert _{{H^1({\varOmega })}^n} =\Vert w_\varepsilon \Vert _{{H^1({\varOmega })}^n} \le c_2c_3\Vert \gamma _0 p_S-\gamma _0 p_PP\Vert _{H^{1/2}({\varGamma })}. \end{aligned}$$

Moreover, by (A.8), we obtain

$$\begin{aligned} \varepsilon \Vert p_PP-p_\varepsilon \Vert ^2_{L^2({\varOmega })}=\varepsilon \Vert r_\varepsilon \Vert ^2_{L^2({\varOmega })}\le c_4\Vert \nabla (p_S-p_PP)\Vert _{{L^2({\varOmega })}^n}\Vert w_\varepsilon \Vert _{{L^2({\varOmega })}^n}. \end{aligned}$$

Hence, if $\gamma _0(p_S-p_PP)=0$, then $p_PP=p_\varepsilon$. $\square$

We show that the sequence $((u_\varepsilon ,p_\varepsilon ))_{\varepsilon >0}$ is bounded in ${H^1({\varOmega })}^n\times ({L^2({\varOmega })}/{{\mathbb {R}}})$. By the reflexivity of ${H^1({\varOmega })}^n\times ({L^2({\varOmega })}/{{\mathbb {R}}})$, the sequence $((u_\varepsilon ,p_\varepsilon ))_{\varepsilon >0}$ has a subsequence converging weakly to somewhere in ${H^1({\varOmega })}^n\times ({L^2({\varOmega })}/{{\mathbb {R}}})$. It is sufficient to check that the limit satisfies (S’). Since the solution of (S’) is unique, the sequence $((u_\varepsilon ,p_\varepsilon ))_{\varepsilon >0}$ converges weakly.

Proof of Theorem 4.2

We take $u_b\in {H^1({\varOmega })}^n,f\in {H^{-1}({\varOmega })}^n$ and $g\in Q^*$ as (A.1) and (A.2) in the proof of Theorem 2.7. We put ${{\tilde{u}}}_\varepsilon :=u_\varepsilon -u_b\in {H^1_0({\varOmega })}^n,{\tilde{p}}_\varepsilon :=p_\varepsilon -p_0\in Q$. Then we obtain

$$\begin{aligned} \left\{ \begin{array}{ll} {\displaystyle \int _{\varOmega }\nabla {\tilde{u}}_\varepsilon :\nabla \varphi +\int _{\varOmega }(\nabla {\tilde{p}}_\varepsilon )\cdot \varphi =\langle f,\varphi \rangle } &{} {\text{ for } \text{ all } } \;\varphi \in {H^1_0({\varOmega })}^n,\\ {\displaystyle \varepsilon \int _{\varOmega }\nabla {\tilde{p}}_\varepsilon \cdot \nabla \psi +\int _{\varOmega }({{\text {div}}}\tilde{u}_\varepsilon )\psi =\varepsilon \langle g,\psi \rangle } &{} {\text{ for } \text{ all } } \;\psi \in Q. \end{array}\right. \end{aligned}$$

(A.9)

Putting $\varphi :={\tilde{u}}_\varepsilon ,\psi :={\tilde{p}}_\varepsilon$ and adding the two equations of (A.9), we get

$$\begin{aligned} \Vert \nabla {\tilde{u}}_\varepsilon \Vert ^2_{{L^2({\varOmega })}^{n\times n}}+\varepsilon \Vert \nabla \tilde{p}_\varepsilon \Vert ^2_{{L^2({\varOmega })}^n}\le & {} \Vert f\Vert _{{H^{-1}({\varOmega })}^n}\Vert \nabla \tilde{u}_\varepsilon \Vert _{{L^2({\varOmega })}^{n\times n}} \\&+\varepsilon \Vert g\Vert _{Q^*}\Vert \nabla \tilde{p}_\varepsilon \Vert _{{L^2({\varOmega })}^n} \end{aligned}$$

since $\int _{\varOmega }(\nabla {\tilde{p}}_\varepsilon )\cdot \tilde{u}_\varepsilon =-\int _{\varOmega }({{\text {div}}}{\tilde{u}}_\varepsilon ){\tilde{p}}_\varepsilon$. Hence,

$$\begin{aligned} (\Vert {\tilde{u}}_\varepsilon \Vert _{{H^1({\varOmega })}^n})_{0<\varepsilon<1} \text{ and } (\Vert \sqrt{\varepsilon }{\tilde{p}}_\varepsilon \Vert _{H^1({\varOmega })})_{0<\varepsilon <1} \text{ are } \text{ bounded. } \end{aligned}$$

(A.10)

Moreover, by Lemma 4.1, we obtain

$$\begin{aligned} \Vert {\tilde{p}}_\varepsilon \Vert _{{L^2({\varOmega })}/{{\mathbb {R}}}}\le {c}(\Vert \nabla \tilde{u}_\varepsilon \Vert _{{L^2({\varOmega })}^{n\times n}}+\Vert f\Vert _{{H^{-1}({\varOmega })}^n}), \end{aligned}$$

i.e., $(\Vert {\tilde{p}}_\varepsilon \Vert _{{L^2({\varOmega })}/{{\mathbb {R}}}})_{0<\varepsilon <1}$ is bounded. By Theorem 3.2, $(\Vert u_\varepsilon \Vert _{{H^1({\varOmega })}^n})_{\varepsilon \ge 1}$ and $(\Vert {\tilde{p}}_\varepsilon \Vert _{{L^2({\varOmega })}/{{\mathbb {R}}}})_{\varepsilon \ge 1}$ are bounded, and thus $(\Vert u_\varepsilon \Vert _{{H^1({\varOmega })}^n})_{\varepsilon >0}$ and $(\Vert \tilde{p}_\varepsilon \Vert _{{L^2({\varOmega })}/{{\mathbb {R}}}})_{\varepsilon >0}$ are bounded.

Since ${H^1({\varOmega })}^n\times ({L^2({\varOmega })}/{{\mathbb {R}}})$ is reflexive and $({\tilde{u}}_\varepsilon ,[\tilde{p}_\varepsilon ])_{0<\varepsilon <1}$ is bounded in ${H^1({\varOmega })}^n\times ({L^2({\varOmega })}/{{\mathbb {R}}})$, there exist $(u,p)\in {H^1({\varOmega })}^n\times ({L^2({\varOmega })}/{{\mathbb {R}}})$ and a subsequence of pairs $({\tilde{u}}_{\varepsilon _k}, {\tilde{p}}_{\varepsilon _k})_{k\in {{\mathbb {N}}}}\subset {H^1_0({\varOmega })}^n\times Q$ such that

$$\begin{aligned} {\tilde{u}}_{\varepsilon _k}\rightharpoonup u~\mathrm{weakly~\text{ in } }{H^1({\varOmega })}^n,~ [{\tilde{p}}_{\varepsilon _k}]\rightharpoonup p~\mathrm{weakly~\text{ in } }{L^2({\varOmega })}/{{\mathbb {R}}}\quad \mathrm{as~}k\rightarrow \infty . \end{aligned}$$

Hence, from (A.9) with $\varepsilon :=\varepsilon _k$, taking $k\rightarrow \infty$, we obtain

$$\begin{aligned} \left\{ \begin{array}{ll}{\displaystyle \int _{\varOmega }\nabla u:\nabla \varphi +\langle \nabla p,\varphi \rangle =\langle f,\varphi \rangle } &{} {\text{ for } \text{ all } } \;\varphi \in {H^1_0({\varOmega })}^n\\ {\displaystyle \int _{\varOmega }({{\text {div}}}u)\psi =0}&{} {\text{ for } \text{ all } } \;\psi \in Q, \end{array}\right. \end{aligned}$$

(A.11)

where we have used that

$$\begin{aligned} |\varepsilon _k\int _{\varOmega }\nabla {\tilde{p}}_{\varepsilon _k}\cdot \nabla \psi |\le & {} \sqrt{\varepsilon _k}\Vert \sqrt{\varepsilon }{\tilde{p}}_\varepsilon \Vert _{H^1({\varOmega })}\Vert \psi \Vert _{H^1({\varOmega })}\rightarrow 0, \\ \int _{\varOmega }\nabla {\tilde{p}}_{\varepsilon _k}\cdot \varphi= & {} -\int _{\varOmega }[\tilde{p}_{\varepsilon _k}]{{\text {div}}}\varphi \rightarrow -\int _{\varOmega }p{{\text {div}}}\varphi =\langle \nabla p,\varphi \rangle \end{aligned}$$

as $k\rightarrow \infty$. By (A.2), the first equation of (A.11) implies that

$$\begin{aligned} \int _{\varOmega }\nabla (u+u_b):\nabla \varphi +\langle \nabla (p+p_0),\varphi \rangle =\int _{\varOmega }F\cdot \varphi \end{aligned}$$

for all $\varphi \in {H^1_0({\varOmega })}^n$. From the second equation of (A.11) and $C^\infty _0({\varOmega })\subset Q$, ${{\text {div}}}(u+u_b)=0$ follows. Hence, we obtain that $(u+u_b,p+[p_0])$ satisfies (S’), i.e., $u_S=u+u_b$ and $p_S=p+[p_0]$. Then we have

$$\begin{aligned} u_{\varepsilon _k}-u_S= & {} u_{\varepsilon _k}-u-u_b=\tilde{u}_{\varepsilon _k}-u_S\rightharpoonup 0 \text{ weakly } \text{ in } {H^1({\varOmega })}^n, \\ {[}p_{\varepsilon _k}{]}-p_S= & {} [p_{\varepsilon _k}-p-p_0]=[\tilde{p}_{\varepsilon _k}]-p\rightharpoonup 0 \text{ weakly } \text{ in } {L^2({\varOmega })}/{{\mathbb {R}}}\end{aligned}$$

as $k\rightarrow \infty$. Since any arbitrarily chosen subsequence of $((u_\varepsilon ,[p_\varepsilon ]))_{0<\varepsilon <1}$ has a subsequence which converges to $(u_S,p_S)$, we can conclude the proof. $\square$

Using Theorem 4.2 and the Rellich–Kondrachov Theorem, it is easy to proveTheorem 4.3.

Proof of Theorem 4.3

We have from (ES’) and (S’) that

$$\begin{aligned} \left\{ \begin{array}{ll} {\displaystyle \int _{\varOmega }\nabla (u_\varepsilon -u_S):\nabla \varphi +\int _{\varOmega }(\nabla (p_\varepsilon -p_S))\cdot \varphi =0 } &{} {\text{ for } \text{ all } } \;\varphi \in {H^1_0({\varOmega })}^n,\\ {\displaystyle \varepsilon \int _{\varOmega }\nabla p_\varepsilon \cdot \nabla \psi +\int _{\varOmega }({{\text {div}}}u_\varepsilon )\psi =\varepsilon \langle G,\psi \rangle } &{} {\text{ for } \text{ all } } \;\psi \in Q. \end{array} \right. \end{aligned}$$

Putting $\varphi :=u_\varepsilon -u_S\in {H^1_0({\varOmega })}^n$ and $\tilde{p}_S:=p_S-p_0\in {H^1({\varOmega })}$, we get

$$\begin{aligned} \begin{aligned} \Vert \nabla (u_\varepsilon -u_S)\Vert ^2_{{L^2({\varOmega })}^{n\times n}}&={\displaystyle -\int _{\varOmega }(\nabla (p_\varepsilon -p_S))\cdot (u_\varepsilon -u_S)}\\&=\displaystyle -\int _{\varOmega }(\nabla (p_\varepsilon -p_0))\cdot (u_\varepsilon -u_S) \\&\quad +\int _{\varOmega }(\nabla (p_S-p_0))\cdot (u_\varepsilon -u_S) \\&={\displaystyle \int _{\varOmega }(p_\varepsilon -p_0){{\text {div}}}(u_\varepsilon -u_S) +\int _{\varOmega }(\nabla {\tilde{p}}_S)\cdot (u_\varepsilon -u_S)}\\&={\displaystyle \int _{\varOmega }(p_\varepsilon -p_0){{\text {div}}}u_\varepsilon +\int _{\varOmega }(\nabla \tilde{p}_S)\cdot (u_\varepsilon -u_S),} \end{aligned} \end{aligned}$$

since $-\int _{\varOmega }(\nabla (p_\varepsilon -p_0)) \cdot (u_\varepsilon -u_S)= \int _{\varOmega }(p_\varepsilon -p_0){{\text {div}}}(u_\varepsilon -u_S)$ and ${{\text {div}}}u_S=0$. Thus,

$$\begin{aligned} \Vert \nabla (u_\varepsilon -u_S)\Vert ^2_{{L^2({\varOmega })}^{n\times n}} =\int _{\varOmega }(p_\varepsilon -p_0){{\text {div}}}u_\varepsilon +\int _{\varOmega }(\nabla \tilde{p}_S)\cdot (u_\varepsilon -u_S). \end{aligned}$$

(A.12)

Putting $\psi :=p_\varepsilon -p_0\in Q$, we have

$$\begin{aligned} \varepsilon \int _{\varOmega }\nabla p_\varepsilon \cdot \nabla (p_\varepsilon -p_0)+\int _{\varOmega }({{\text {div}}}u_\varepsilon )(p_\varepsilon -p_0) =\varepsilon \langle G,p_\varepsilon -p_0\rangle . \end{aligned}$$

Hence,

$$\begin{aligned} \varepsilon \Vert \nabla (p_\varepsilon -p_0)\Vert ^2_{{L^2({\varOmega })}^n}&=-\varepsilon \int _{\varOmega }\nabla (p_\varepsilon -p_0)\cdot \nabla p_0 \nonumber \\&\quad -\int _{\varOmega }(p_\varepsilon -p_0){{\text {div}}}u_\varepsilon +\varepsilon \langle G,p_\varepsilon -p_0\rangle . \end{aligned}$$

(A.13)

Together with (A.12) and (A.13), we obtain

$$\begin{aligned} \begin{aligned}&\Vert \nabla (u_\varepsilon -u_S)\Vert ^2_{{L^2({\varOmega })}^{n\times n}}+ \varepsilon \Vert \nabla (p_\varepsilon -p_0)\Vert ^2_{{L^2({\varOmega })}^n}\\&\quad ={\displaystyle \int _{\varOmega }\nabla {\tilde{p}}_S\cdot (u_\varepsilon -u_S) -\varepsilon \int _{\varOmega }\nabla (p_\varepsilon -p_0)\cdot \nabla p_0 +\varepsilon \langle G,p_\varepsilon -p_0\rangle }\\&\quad \le \Vert \nabla {\tilde{p}}_S\Vert _{{L^2({\varOmega })}^n}\Vert u_\varepsilon -u_S\Vert _{{L^2({\varOmega })}^n} \\&\qquad +\varepsilon (\Vert \nabla p_0\Vert _{{L^2({\varOmega })}^n}+\Vert G\Vert _{Q^*})\Vert \nabla (p_\varepsilon -p_0)\Vert _{{L^2({\varOmega })}^n}. \end{aligned} \end{aligned}$$

By Theorem 4.2 and the Rellich–Kondrachov Theorem, there exists a sequence $(\varepsilon _k)_{k\in {{\mathbb {N}}}}\subset {{\mathbb {R}}}$ such that

$$\begin{aligned} u_{\varepsilon _k}\rightarrow u_S \text{ strongly } \text{ in } {L^2({\varOmega })}^n \quad \text{ as } k\rightarrow \infty . \end{aligned}$$

Therefore, by (A.10),

$$\begin{aligned} \begin{aligned} \Vert \nabla (u_{\varepsilon _k}-u_S)\Vert ^2_{{L^2({\varOmega })}^{n\times n}}&\le \Vert \nabla \tilde{p}_S\Vert _{{L^2({\varOmega })}^n}\Vert u_{\varepsilon _k}-u_S\Vert _{{L^2({\varOmega })}^n} \\&\quad +\varepsilon _k(\Vert \nabla {\tilde{p}}_0\Vert _{{L^2({\varOmega })}^n}+\Vert G\Vert _{Q^*})\Vert \nabla (p_{\varepsilon _k}-p_0)\Vert _{{L^2({\varOmega })}^n}\\&\rightarrow 0 \end{aligned} \end{aligned}$$

as $k\rightarrow \infty$. This implies that

$$\begin{aligned} \Vert [p_{\varepsilon _k}]-p_S\Vert _{{L^2({\varOmega })}/{{\mathbb {R}}}}= & {} \Vert p_{\varepsilon _k}-p_S\Vert _{{L^2({\varOmega })}/{{\mathbb {R}}}} \\\le & {} c\Vert \nabla (u_{\varepsilon _k}-u_S)\Vert _{{L^2({\varOmega })}^{n\times n}} \rightarrow 0 \text{ as } k\rightarrow \infty \end{aligned}$$

by Lemma 4.1. Since any arbitrarily chosen subsequence of $((u_\varepsilon ,[p_\varepsilon ]))_{0<\varepsilon <1}$ has a subsequence which converges to $(u_S,p_S)$, we can conclude the proof. $\square$