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

TL;DR

This paper extends the analysis of an $oldsymbol{ ext{ε}}$-Stokes projection method, connecting Stokes and pressure-Poisson problems, providing convergence, error estimates, and numerical validation for various boundary conditions.

Contribution

It generalizes the $oldsymbol{ ext{ε}}$-Stokes approach to Neumann and mixed boundary conditions, offering new error estimates and insights into asymptotic behavior.

Findings

Solutions converge to Stokes and pressure-Poisson solutions as ε varies.

Error estimates are optimal and improved with Neumann boundary conditions.

Numerical examples confirm theoretical error bounds and asymptotic structure.

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 K. Matsui and A. Muntean (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.

Equations

$$\displaystyle\left\{\begin{array}[]{ll}-\Delta u_{\mbox{\tiny$S$}}+\nabla p_{\mbox{\tiny$S$}}=F&\mbox{in }\Omega,\\ {\operatorname{div}}u_{\mbox{\tiny$S$}}=0&\mbox{in }\Omega,\\ u_{\mbox{\tiny$S$}}=u_{b}&\mbox{on }\Gamma,\end{array}\right.$$

$$\displaystyle{\operatorname{div}}F={\operatorname{div}}(-\Delta u_{\mbox{\tiny$S$}}+\nabla p_{\mbox{\tiny$S$}})=-\Delta({\operatorname{div}}u_{\mbox{\tiny$S$}})+\Delta p_{\mbox{\tiny$S$}}=\Delta p_{\mbox{\tiny$S$}}.$$

$$\displaystyle\left\{\begin{array}[]{ll}-\Delta u_{\mbox{\tiny$P\!P$}}+\nabla p_{\mbox{\tiny$P\!P$}}=F&\mbox{in }\Omega,\\ -\Delta p_{\mbox{\tiny$P\!P$}}=-{\operatorname{div}}F&\mbox{in }\Omega,\\ u_{\mbox{\tiny$P\!P$}}=u_{b}&\mbox{on }\Gamma,\\ +\mbox{boundary condition for }p_{\mbox{\tiny$P\!P$}}.&\end{array}\right.$$

$$\displaystyle\left\{\begin{array}[]{ll}-\Delta u_{\varepsilon}+\nabla p_{\varepsilon}=F&\mbox{in }\Omega,\\ -\varepsilon\Delta p_{\varepsilon}+{\operatorname{div}}u_{\varepsilon}=-\varepsilon{\operatorname{div}}F&\mbox{in }\Omega,\\ u_{\varepsilon}=u_{b}&\mbox{on }\Gamma,\\ +\mbox{boundary condition for }p_{\varepsilon}.&\end{array}\right.$$

$$\begin{array}[]{rrl}\|u_{\mbox{\tiny$S$}}-u_{\mbox{\tiny$P\!P$}}\|_{{H^{1}(\Omega)}^{n}}&\leq&c\|\gamma_{0}p_{\mbox{\tiny$S$}}-\gamma_{0}p_{\mbox{\tiny$P\!P$}}\|_{H^{1/2}(\Gamma)},\\ \|u_{\mbox{\tiny$S$}}-u_{\varepsilon}\|_{{H^{1}(\Omega)}^{n}}&\leq&c\|\gamma_{0}p_{\mbox{\tiny$S$}}-\gamma_{0}p_{\mbox{\tiny$P\!P$}}\|_{H^{1/2}(\Gamma)},\end{array}$$

$$(PP)\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$|\Gamma \setminus ({\Gamma }_D\cup {\Gamma }_N)|=0,|{\Gamma }_D|>0,{\Gamma }_D\cap {\Gamma }_N=\varnothing ,$$

$$\displaystyle{\displaystyle\frac{\partial p_{\mbox{\tiny$P\!P$}}}{\partial\nu}=g_{b}}\mbox{ on }\Gamma,\quad{\displaystyle\frac{\partial p_{\varepsilon}}{\partial\nu}=g_{b}}\mbox{ on }\Gamma,$$

$$\displaystyle\left\{\begin{array}[]{ll}p_{\mbox{\tiny$P\!P$}}=p_{b}&\mbox{on }\Gamma_{D},\\[8.0pt] {\displaystyle\frac{\partial p_{\mbox{\tiny$P\!P$}}}{\partial\nu}=g_{b}}&\mbox{on }\Gamma_{N},\end{array}\right.\quad\left\{\begin{array}[]{ll}p_{\varepsilon}=p_{b}&\mbox{on }\Gamma_{D},\\[8.0pt] {\displaystyle\frac{\partial p_{\varepsilon}}{\partial\nu}=g_{b}}&\mbox{on }\Gamma_{N},\end{array}\right.$$

$$\begin{array}[]{ll}C^{\infty}_{0}(\Omega)^{n}&{\displaystyle:=\{\varphi\in C^{\infty}(\Omega)^{n}\,|\,\mbox{supp}(\varphi)\subset\Omega\},}\\[8.0pt] {L^{2}(\Omega)}/{\mathbb{R}}&:={\displaystyle\left\{\psi\in{L^{2}(\Omega)}\,\middle|\,\int_{\Omega}\psi=0\right\},}\\[8.0pt] {H^{1}(\Omega)}/{\mathbb{R}}&:={H^{1}(\Omega)}\cap({L^{2}(\Omega)}/{\mathbb{R}}),\\[8.0pt] {H^{1}_{0,D}(\Omega)}&:={\displaystyle\{\psi\in{H^{1}(\Omega)}\mid\psi|_{\Gamma_{D}}=0\}.}\\[8.0pt] \end{array}$$

$$\Vert f{\Vert }_{{H^{-1}(\Omega )}^m}:=\underset{\phi \in S_m}{\sup }⟨f,\phi ⟩\text{mbox}forf\in {H^{-1}(\Omega )}^m,$$

$$S_m:=\{\phi \in {H_0^1(\Omega )}^m|\Vert \nabla \phi {\Vert }_{{L^2(\Omega )}^{n\times m}}=1\}.$$

$$\Vert f{\Vert }_{Q^{\ast }}:=\underset{\psi \in S_Q}{\sup }⟨f,\psi ⟩$$

$$S_Q:=\{\psi \in Q|\Vert \nabla \psi {\Vert }_{{L^2(\Omega )}^n}=1\}.$$

$$\begin{array}[]{rl}[p]&:={\displaystyle p-\frac{1}{|\Omega|}\int_{\Omega}p,}\\[8.0pt] \|p\|_{{L^{2}(\Omega)}/{\mathbb{R}}}&:={\displaystyle\inf_{a\in{\mathbb{R}}}\|p-a\|_{L^{2}(\Omega)}=\|[p]\|_{L^{2}(\Omega)},}\\[8.0pt] \langle\nabla p,\varphi\rangle&:={\displaystyle-\int_{\Omega}p\,{\operatorname{div}}\varphi}\\[8.0pt] {\displaystyle\nabla u:\nabla\varphi}&:={\displaystyle\sum^{n}_{i=1}\frac{\partial u}{\partial x_{i}}\cdot\frac{\partial\varphi}{\partial x_{i}}}\end{array}$$

$${\int }_{\Omega }u\cdot \nabla \omega +{\int }_{\Omega }(\mathrm{div}u)\omega ={\int }_{\Gamma }(u\cdot \nu )\omega .$$

$$\Vert f{\Vert }_{L^2(\Omega )}\le c(\Vert f{\Vert }_{H^{-1}(\Omega )}+\Vert \nabla f{\Vert }_{{H^{-1}(\Omega )}^n})$$

$$\Vert f{\Vert }_{L^2(\Omega )/\mathbb{R}}\le c\Vert \nabla f{\Vert }_{{H^{-1}(\Omega )}^n}$$

$$\Vert \psi {\Vert }_{L^2(\Omega )}\le c\Vert \nabla \psi {\Vert }_{{L^2(\Omega )}^n}$$

$$F\in {L^2(\Omega )}^n,u_b\in H^{1/2}(\Gamma ),{\int }_{\Gamma }{u}_b\cdot \nu =0,$$

$$g_b\in L^2(\Gamma ),\mathrm{div}F\in L^2(\Omega ),$$

$${\int }_{\Gamma }{g}_b={\int }_{\Omega }\mathrm{div}F,$$

$$p_b\in H^1(\Omega ).$$

$$\int_{\Omega}F\cdot\varphi=-\int_{\Gamma}\frac{\partial u_{\mbox{\tiny$S$}}}{\partial\nu}\cdot\varphi+\int_{\Omega}\nabla u_{\mbox{\tiny$S$}}:\nabla\varphi+\int_{\Omega}\nabla p_{\mbox{\tiny$S$}}\cdot\varphi=\int_{\Omega}\nabla u_{\mbox{\tiny$S$}}:\nabla\varphi+\int_{\Omega}\nabla p_{\mbox{\tiny$S$}}\cdot\varphi.$$

$$\displaystyle\left\{\begin{array}[]{ll}{\displaystyle\int_{\Omega}\nabla u_{\mbox{\tiny$S$}}:\nabla\varphi+\langle\nabla p_{\mbox{\tiny$S$}},\varphi\rangle=\int_{\Omega}F\cdot\varphi}&\mbox{for all }\varphi\in{H^{1}_{0}(\Omega)}^{n},\\[8.0pt] {\operatorname{div}}u_{\mbox{\tiny$S$}}=0&\mbox{in }{L^{2}(\Omega)},\\ u_{\mbox{\tiny$S$}}=u_{b}&\mbox{in }H^{1/2}(\Gamma)^{n}.\end{array}\right.$$

$$\begin{array}[]{ll}{\displaystyle-\int_{\Omega}({\operatorname{div}}F)\psi}&{\displaystyle=-\int_{\Omega}(\Delta p_{\mbox{\tiny$P\!P$}})\psi}={\displaystyle-\int_{\Gamma}\frac{\partial p_{\mbox{\tiny$P\!P$}}}{\partial\nu}\psi+\int_{\Omega}\nabla p_{\mbox{\tiny$P\!P$}}\cdot\nabla\psi}\\[8.0pt] &{\displaystyle=-\int_{\Gamma}g_{b}\psi+\int_{\Omega}\nabla p_{\mbox{\tiny$P\!P$}}\cdot\nabla\psi}.\end{array}$$

$$\int_{\Omega}\nabla p_{\mbox{\tiny$P\!P$}}\cdot\nabla\psi=\int_{\Gamma}g_{b}\psi-\int_{\Omega}({\operatorname{div}}F)\psi.$$

$$\displaystyle\left\{\begin{array}[]{ll}{\displaystyle\int_{\Omega}\nabla u_{\mbox{\tiny$P\!P$}}:\nabla\varphi+\int_{\Omega}\nabla p_{\mbox{\tiny$P\!P$}}\cdot\varphi=\int_{\Omega}F\cdot\varphi}&\mbox{for all }\varphi\in{H^{1}_{0}(\Omega)}^{n},\\[8.0pt] {\displaystyle\int_{\Omega}\nabla p_{\mbox{\tiny$P\!P$}}\cdot\nabla\psi=\langle G_{1},\psi\rangle}&\mbox{for all }\psi\in{H^{1}(\Omega)}/{\mathbb{R}},\\ u_{\mbox{\tiny$P\!P$}}=u_{b}&\mbox{in }H^{1/2}(\Gamma)^{n},\end{array}\right.$$

$$\displaystyle\left\{\begin{array}[]{ll}{\displaystyle\int_{\Omega}\nabla u_{\varepsilon}:\nabla\varphi+\int_{\Omega}\nabla p_{\varepsilon}\cdot\varphi=\int_{\Omega}F\cdot\varphi}&\mbox{for all }\varphi\in{H^{1}_{0}(\Omega)}^{n},\\[8.0pt] {\displaystyle\varepsilon\int_{\Omega}\nabla p_{\varepsilon}\cdot\nabla\psi+\int_{\Omega}({\operatorname{div}}u_{\varepsilon})\psi=\varepsilon\langle G_{1},\psi\rangle}&\mbox{for all }\psi\in{H^{1}(\Omega)}/{\mathbb{R}},\\ u_{\mbox{\tiny$S$}}=u_{b}&\mbox{in }H^{1/2}(\Gamma)^{n}.\end{array}\right.$$

Full text

1 Introduction

Let $\Omega$ be a bounded domain in ${\mathbb{R}}^{n}(n\geq 2,n\in{\mathbb{N}})$ with Lipschitz continuous boundary $\Gamma$ and let $F:\Omega\rightarrow{\mathbb{R}}^{n}$ be a given applied force field and $u_{b}:\Gamma\rightarrow{\mathbb{R}}^{n}$ be a given Dirichlet boundary data satisfying $\int_{\Gamma}u_{b}\cdot\nu=0$, where $\nu$ is the unit outward normal vector on $\Gamma$. A strong form of the Stokes problem is given as follows. Find u_{\mbox{\tinyS}}:\Omega\rightarrow{\mathbb{R}}^{n} and p_{\mbox{\tinyS}}:\Omega\rightarrow{\mathbb{R}} such that

[TABLE]

where u_{\mbox{\tinyS}} and p_{\mbox{\tinyS}} are the velocity and the pressure of the flow governed by (S), respectively. We refer to Temam for the details on the Stokes problem (i.e., physical background and corresponding mathematical analysis). Taking the divergence of the first equation, we obtain

[TABLE]

This equation is often called the pressure-Poisson equation and is used in numerical schemes such as MAC (marker and cell), SMAC (simplified MAC) and the projection methods (see, e.g., Amsden_Harlow ; Chorin68 ; Cummins_Rudman ; Guermond ; mac1 ; Kim_Moin ; mac2 ; Perot ). Based on the above, we consider a similar problem. Find u_{\mbox{\tinyP!P}}:\Omega\rightarrow{\mathbb{R}}^{n} and p_{\mbox{\tinyP!P}}:\Omega\rightarrow{\mathbb{R}} satisfying

[TABLE]

We call this problem the pressure-Poisson problem. The idea of using (1.5) instead of {\operatorname{div}}u_{\mbox{\tinyS}}=0 is useful for calculating the pressure numerically in the Navier–Stokes equation. For example, this idea is used in MAC, SMAC and projection methods. The Dirichlet boundary condition for the pressure is used in an outflow boundary Chan_Street ; Viecelli . See also Conca_etc94 ; Conca_etc95 ; Marusic .

We introduce an “interpolation” between problems (S) and (PP). For $\varepsilon>0$, find $u_{\varepsilon}:\Omega\rightarrow{\mathbb{R}}^{n}$ and $p_{\varepsilon}:\Omega\rightarrow{\mathbb{R}}$ such that

[TABLE]

This problem is called the $\varepsilon$-Stokes problem (ES) in prev . In Douglas_Wang ; Glowinski ; Hughes , the authors treat this problem as an approximation of the Stokes problem to avoid numerical instabilities. The $\varepsilon$-Stokes problem approximates the Stokes problem (S) as $\varepsilon\rightarrow 0$ and the pressure-Poisson problem (PP) as $\varepsilon\rightarrow\infty$ (Fig. 1). It is shown in prev that if p_{\mbox{\tinyS}}\in{H^{1}(\Omega)} then there exists a constant $c>0$ independent of $\varepsilon$ such that

[TABLE]

where $\gamma_{0}\in B({H^{1}(\Omega)},H^{1/2}(\Gamma))$ is the standard trace operator Girault . From the first inequality, if we have a good prediction value for pressure on $\Gamma$, then u_{\mbox{\tinyP!P}} is a good approximation of u_{\mbox{\tinyS}}. Moreover, $u_{\varepsilon}$ is also a good approximation of u_{\mbox{\tinyS}} from the second inequality.

Next we specify the boundary conditions for p_{\mbox{\tinyP!P}} and $p_{\varepsilon}$. We assume that the boundary $\Gamma$ is a union of two open subsets $\Gamma_{D}$ and $\Gamma_{N}$ such that

[TABLE]

and number of connected components of $\Gamma_{D}$ and $\Gamma_{N}$ with respect to the relative topology of $\Gamma$ are finite. We consider a Neumann boundary condition (1.16) and a mixed boundary condition (1.21),

[TABLE]

[TABLE]

where $p_{b}:\Gamma_{D}\rightarrow{\mathbb{R}}$ and $g_{b}=\Gamma\rightarrow{\mathbb{R}}$ satisfying $\int_{\Gamma}g_{b}=\int_{\Gamma}{\operatorname{div}}F$ are given boundary data.

In prev , the authors impose Dirichlet boundary conditions for p_{\mbox{\tinyP!P}} and $p_{\varepsilon}$ (i.e., (1.21) with $\Gamma_{D}=\Gamma$ and $\Gamma_{N}=\emptyset$.) For such boundary conditions, they introduce a weak solution $(u_{\varepsilon},p_{\varepsilon})$ to the $\varepsilon$-Stokes problem (ES) and prove that $(u_{\varepsilon},p_{\varepsilon})$ strongly converges in ${H^{1}(\Omega)}^{n}\times{H^{1}(\Omega)}$ to a weak solution to the pressure-Poisson problem (PP) as $\varepsilon\rightarrow\infty$ and weakly converges in ${H^{1}_{0}(\Omega)}^{n}\times({L^{2}(\Omega)}/{\mathbb{R}})$ to a weak solution (u_{\mbox{\tinyS}},p_{\mbox{\tinyS}}) to the Stokes problem (S) as $\varepsilon\rightarrow 0$. Moreover, if p_{\mbox{\tinyS}}\in{H^{1}(\Omega)}, then strong convergence of $(u_{\varepsilon},p_{\varepsilon})$ to (u_{\mbox{\tinyS}},p_{\mbox{\tinyS}}) as $\varepsilon\rightarrow 0$ holds.

In this paper, we generalize the Dirichlet boundary condition of p_{\mbox{\tinyP!P}} and $p_{\varepsilon}$ to both the Neumann boundary condition (1.16) and the mixed boundary condition (1.21), and prove the corresponding convergence result (see Theorem 3.1, 4.2 and 4.3). Since the mixed boundary condition for pressure often appears in engineering problems, this generalization of the boundary conditions for pressure is important. In addition, for the Neumann boundary condition, we estimate the error between the weak solutions to (ES) and (S) provided p_{\mbox{\tinyS}}\in{H^{1}(\Omega)}. We also give an asymptotic expansion for the weak solution to (ES). We furthermore check this convergence result using numerical computations.

The organization of this paper is as follows. In Section 2 we introduce the notation used in this work and the weak form of these problems. We also prove the well-posedness of the problems (PP) and (ES) and show some their properties. In Section 3 we study that the solution to (ES) converges to the solution to (PP) in the strong topology as $\varepsilon\rightarrow\infty$. We also explore here the structure of the regular perturbation asymptotics. Section 4 is devoted to proving that the solution to (ES) converges to the solution to (S) in the weak and strong topology as $\varepsilon\rightarrow 0$. Finally, in Section 5, we show several numerical examples of these problems. The numerical errors between the problems (ES) and (PP), and between the problems (ES) and (S) using the P2/P1 finite element method. Proofs for several theorems which are similar to ones in prev are described.

2 Well-posedness

In this section, we introduce the notation and the weak form of the problems (S), (PP) and (ES), and prove their well-posedness. We give estimates between these solutions by using a pressure error on the boundary $\Gamma$.

2.1 Notation

We set

[TABLE]

For $m\in{\mathbb{N}}$, ${H^{-1}(\Omega)}^{m}:=({H^{1}_{0}(\Omega)}^{m})^{*}$ is equipped with the dual norm

[TABLE]

where

[TABLE]

Let $Q\subset{H^{1}(\Omega)}$ be a closed subspace such that there exists a constant $c>0$ for which $\|q\|_{L^{2}(\Omega)}\leq c\|\nabla q\|_{{L^{2}(\Omega)}^{n}}$ for all $q\in Q$. The dual space $Q^{*}$ is equipped with the norm

[TABLE]

for $f\in Q^{*}$, where

[TABLE]

We define

[TABLE]

for all $p\in{L^{2}(\Omega)},u,\varphi\in{H^{1}_{0}(\Omega)}^{n}$, where $|\Omega|$ is the volume of $\Omega$.

2.2 Preliminary results

Let $\gamma_{0}\in B({H^{1}(\Omega)},H^{1/2}(\Gamma))$ be the standard trace operator. The trace operator $\gamma_{0}$ is surjective and satisfies $\mbox{Ker}(\gamma_{0})={H^{1}_{0}(\Omega)}$ (Girault, , Theorem 1.5). Let $\nu$ be the unit outward normal for $\Gamma$. Since $\nu$ is a unit vector, ${H^{1}(\Omega)}^{3}\ni u\mapsto u\cdot\nu:=(\gamma_{0}u)\cdot\nu\in H^{1/2}(\Gamma)$ is a linear continuous map. For all $u\in{H^{1}(\Omega)}^{3}$ and $\omega\in{H^{1}(\Omega)}$, the following Gauss divergence formula holds:

[TABLE]

We recall the following four embedding theorems which plays an important role in the proof of the existence of pressure solutions to the Stokes problem. For the proof of Theorem 2.1, see (Necas, , Lemma 7.1) and (Duvaut, , Theorem 3.2 and Remark 3.1).

Theorem 2.1

There exists a constant $c>0$ such that

[TABLE]

for all $f\in{L^{2}(\Omega)}$.

The following result follows from Theorem 2.1.

Theorem 2.2

(Girault, , Corollary 2.1, 2∘)* There exists a constant $c>0$ such that*

[TABLE]

for all $f\in{L^{2}(\Omega)}$.

The following two embedding theorems are often called the Poincaré inequality.

Theorem 2.3

(Necas, , Theorem 7.8)* There exists a constant $c>0$ such that*

[TABLE]

for all $\psi\in{H^{1}(\Omega)}/{\mathbb{R}}$.

Theorem 2.4

(Girault, , Lemma 3.1)* There exists a constant $c>0$ such that*

[TABLE]

for all $\psi\in{H^{1}_{0,D}(\Omega)}$.

2.3 Weak formulations of the problems (PP), (S) and (ES)

We assume the following conditions for $F,u_{b}$ and $g_{b}$:

[TABLE]

[TABLE]

[TABLE]

[TABLE]

We start by defining the weak solution to (S). For all $\varphi\in{H^{1}_{0}(\Omega)}^{n}$, we obtain from the first equation of (S) that

[TABLE]

Using this expression, the weak form of the Stokes problem becomes as follows: Find u_{\mbox{\tinyS}}\in{H^{1}(\Omega)}^{n} and p_{\mbox{\tinyS}}\in{L^{2}(\Omega)}/{\mathbb{R}} such that

[TABLE]

Next, we define the weak formulations of (PP) and (ES) first for the Neumann boundary condition (1.16) and them for the mixed boundary condition (1.21). After that, we define generalized weak formulations for (PP) and (ES) which cover both cases.

First, we apply the Neumann boundary condition (1.16) for (PP) and (ES). We take a test function $\psi\in{H^{1}(\Omega)}$. From the second equation of (PP), we obtain

[TABLE]

Hence,

[TABLE]

We note that $\int_{\Gamma}g_{b}\psi-\int_{\Omega}({\operatorname{div}}F)\psi=\int_{\Gamma}g_{b}[\psi]-\int_{\Omega}({\operatorname{div}}F)[\psi]$ for all $\psi\in{H^{1}(\Omega)}$ by (2.24). Therefore, the weak form of the pressure-Poisson problem with the Neumann boundary condition (1.16) becomes as follows. Find u_{\mbox{\tinyP!P}}\in{H^{1}(\Omega)}^{n} and p_{\mbox{\tinyP!P}}\in{H^{1}(\Omega)}/{\mathbb{R}} such that

[TABLE]

where $G_{1}\in{H^{1}(\Omega)}^{*}$ such that $\langle G_{1},\psi\rangle=\int_{\Gamma}g_{b}\psi-\int_{\Omega}({\operatorname{div}}F)\psi~{}(\psi\in{H^{1}(\Omega)})$.

The weak form of (ES) with the Neumann boundary condition can be defined similarly to that of (PP). Find $u_{\varepsilon}\in{H^{1}(\Omega)}^{n}$ and $p_{\varepsilon}\in{H^{1}(\Omega)}/{\mathbb{R}}$ such that

[TABLE]

Secondly, we apply the mixed boundary condition (1.21) for (PP) and (ES). We take a test function $\psi\in{H^{1}_{0,D}(\Omega)}$. From the second equation of (PP), we obtain

[TABLE]

Hence,

[TABLE]

The weak form of the pressure-Poisson problem with the mixed boundary condition (1.21) becomes as follows. Find u_{\mbox{\tinyP!P}}\in{H^{1}(\Omega)}^{n} and p_{\mbox{\tinyP!P}}\in{H^{1}(\Omega)} such that

[TABLE]

where $G_{2}\in{H^{1}_{0,D}(\Omega)}^{*}$ such that

[TABLE]

for $\psi\in{H^{1}_{0,D}(\Omega)}$. The weak form of (ES) with the mixed boundary condition (1.21) can be defined similarly to that of (PP). It reads as follows. Find $u_{\varepsilon}\in{H^{1}(\Omega)}^{n}$ and $p_{\varepsilon}\in{H^{1}(\Omega)}$ such that

[TABLE]

Finally, we generalize (PP1) and (PP2) to an abstract pressure-Poisson problem. Let $Q\subset{H^{1}(\Omega)}$ be a closed subspace as defined in Section 2.1. Find u_{\mbox{\tinyP!P}}\in{H^{1}(\Omega)}^{n} and p_{\mbox{\tinyP!P}}\in Q such that

[TABLE]

with $G\in Q^{*}$. Indeed, by Theorem 2.3 and 2.4, we obtain (PP1) from (PP’) by putting $Q:={H^{1}(\Omega)}/{\mathbb{R}}$ and $G:=G_{1}$. Similarly, we obtain (PP2) from (PP’) by putting $Q:={H^{1}_{0,D}(\Omega)}$ and $G:=G_{2}$.

We generalize (ES1) and (ES2) to an abstract $\varepsilon$-Stokes problem. Find $u_{\varepsilon}\in{H^{1}(\Omega)}^{n}$ and $p_{\varepsilon}\in Q$ such that

[TABLE]

Indeed, by Theorem 2.3, 2.4, we obtain (ES1) from (ES’) by putting $Q:={H^{1}(\Omega)}/{\mathbb{R}}$ and $G:=G_{1}$. Similarly, we also obtain (ES2) from (ES’) by putting $Q:={H^{1}_{0,D}(\Omega)}$ and $G:=G_{2}$.

2.4 Well-posedness of (S’), (PP’) and (ES’)

We show the well-posedness of problems (S’), (PP’) and (ES’) in Theorem 2.5, 2.6 and 2.7.

Theorem 2.5

Under the condition (2.22), there exists a unique solution

(u_{\mbox{\tinyS}},p_{\mbox{\tinyS}})\in{H^{1}(\Omega)}^{n}\times({L^{2}(\Omega)}/{\mathbb{R}})* satisfying (S’).*

See (Temam, , Theorem 2.4 and Remark 2.5) for the proof.

Theorem 2.6

Under the condition (2.22) and (2.25), for $G\in Q^{*}$, there exists a unique solution (u_{\mbox{\tinyP!P}},p_{\mbox{\tinyP!P}})\in{H^{1}(\Omega)}^{n}\times Q satisfying (PP’).

Proof. Using the Lax–Milgram theorem, since $Q\times Q\ni(p,\psi)\mapsto\int_{\Omega}\nabla p\cdot\nabla\psi\in{\mathbb{R}}$ is a continuous and coercive bilinear form, p_{\mbox{\tinyP!P}}\in{H^{1}(\Omega)} is uniquely determined from the second and fourth equations of (PP’). Then u_{\mbox{\tinyP!P}}\in{H^{1}(\Omega)}^{n} is also uniquely determined from the first and third equations, again using the Lax–Milgram theorem. ∎

Theorem 2.7

Under the condition (2.22) and (2.25), for $\varepsilon>0$ and $G\in Q^{*}$, there exists a unique solution $(u_{\varepsilon},p_{\varepsilon})\in{H^{1}(\Omega)}^{n}\times{H^{1}(\Omega)}$ satisfying (ES’).

This is a generalization of Theorem 2.6 in prev . See Appendix Appendix for the proof.

From now on, let the solutions of (S’), (PP’) and (ES’) be denoted by (u_{\mbox{\tinyS}},p_{\mbox{\tinyS}}),(u_{\mbox{\tinyP!P}},p_{\mbox{\tinyP!P}}) and $(u_{\varepsilon},p_{\varepsilon})$, respectively. We show their properties in connection with a pressure error on the boundary $\Gamma$.

Proposition 2.8

*Suppose that p_{\mbox{\tinyS}}\in{H^{1}(\Omega)}, ${H^{1}_{0}(\Omega)}\subset Q$ and $\langle G,\psi\rangle=$

$-\int_{\Omega}({\operatorname{div}}F)\psi$ for all $\psi\in{H^{1}_{0}(\Omega)}$. Then there exists a constant $c>0$ independent of $\varepsilon$ such that*

[TABLE]

In particular, if \gamma_{0}p_{\mbox{\tinyS}}=\gamma_{0}p_{\mbox{\tinyP!P}}, then (u_{\mbox{\tinyS}},p_{\mbox{\tinyS}})=(u_{\mbox{\tinyP!P}},p_{\mbox{\tinyP!P}})=(u_{\varepsilon},p_{\varepsilon}) holds for all $\varepsilon>0$.

This is a generalization of Proposition 2.7 in prev . See Appendix A for the proof.

Since ${H^{1}_{0}(\Omega)}\not\subset{H^{1}(\Omega)}/{\mathbb{R}}$, Proposition 2.8 does not apply directly for the case of the Neumann boundary condition (1.16). However, we add natural assumptions, then it leads to (2.60).

Proposition 2.9

Suppose that p_{\mbox{\tinyS}}\in{H^{1}(\Omega)}, $Q={H^{1}(\Omega)}/{\mathbb{R}}$. If $G\in Q^{*}=({H^{1}(\Omega)}/{\mathbb{R}})^{*}$ is such that $G\in{H^{1}(\Omega)}^{*}$ and $\langle G,\psi\rangle=\int_{\Gamma}g_{b}\psi-\int_{\Omega}({\operatorname{div}}F)\psi$ for all $\psi\in{H^{1}(\Omega)}$, then we have (2.60).

Proof. Since $G\in{H^{1}(\Omega)}^{*}$ satisfies $\langle G,\psi\rangle=\int_{\Gamma}g_{b}\psi-\int_{\Omega}({\operatorname{div}}F)\psi$ for all $\psi\in{H^{1}(\Omega)}$, it holds that

[TABLE]

for all $\psi\in{H^{1}_{0}(\Omega)}$ from the second equation of (PP’). Hence, it leads the second equation of (.10). Using the proof of Proposition 2.8, we obtain (2.60). ∎

3 Links between (ES) and (PP)

as guaranteed by Theorem 2.5, 2.6 and 2.7. In this section, we show that $(u_{\varepsilon},p_{\varepsilon})$ converges to (u_{\mbox{\tinyP!P}},p_{\mbox{\tinyP!P}}) strongly in ${H^{1}(\Omega)}^{n}\times{H^{1}(\Omega)}$ as $\varepsilon\rightarrow\infty$. We also treat the case of the regular perturbation asymptotics by exploring the structure of the lower order terms and their effect on the convergence rate.

3.1 Convergence as $\varepsilon\rightarrow\infty$

We use the following Lemma 3.1 for the proofs of the theorems in this section.

Lemma 3.1

Let $h\in Q^{*}$ and $(v_{\varepsilon},q_{\varepsilon})\in{H^{1}_{0}(\Omega)}^{n}\times Q$ satisfy

[TABLE]

for an arbitrarily fixed $\varepsilon>0$. Then there exists a constant $c>0$ such that

[TABLE]

Proof. Putting $\varphi:=v_{\varepsilon}$ and $\psi:=q_{\varepsilon}$ and adding two equations of (3.63), we obtain

[TABLE]

where we have used $\int_{\Omega}\nabla q_{\varepsilon}\cdot v_{\varepsilon}=-\int_{\Omega}({\operatorname{div}}v_{\varepsilon})q_{\varepsilon}$. Thus

[TABLE]

In addition, from the first equation of (3.63) by putting $\varphi:=v_{\varepsilon}$, we have

[TABLE]

and then

[TABLE]

∎

Using Lemma 3.1, we obtain Theorem 3.2.

Theorem 3.2

There exists a constant $c>0$ independent of $\varepsilon>0$ such that

[TABLE]

for all $\varepsilon>0$. In particular, we have

[TABLE]

Proof. Combining (PP’) and (ES’), we obtain

[TABLE]

where v_{\varepsilon}:=u_{\varepsilon}-u_{\mbox{\tinyP!P}},q_{\varepsilon}:=p_{\varepsilon}-p_{\mbox{\tinyP!P}} and h:={\operatorname{div}}u_{\mbox{\tinyP!P}}. By Lemma 3.1, we conclude the proof. ∎

Corollary 3.3

If u_{\mbox{\tinyP!P}} satisfies {\operatorname{div}}u_{\mbox{\tinyP!P}}=0, then u_{\varepsilon}=u_{\mbox{\tinyP!P}} and p_{\varepsilon}=p_{\mbox{\tinyP!P}} hold for all $\varepsilon>0$. Furthermore, u_{\mbox{\tinyS}}=u_{\varepsilon}=u_{\mbox{\tinyP!P}} and p_{\mbox{\tinyS}}=[p_{\varepsilon}]=[p_{\mbox{\tinyP!P}}] hold for all $\varepsilon>0$.

3.2 Regular Perturbation Asymptotics

By Theorem 3.2, we have that \|\varepsilon(u_{\varepsilon}-u_{\mbox{\tinyP!P}})\|_{{H^{1}(\Omega)}^{n}}\leq c and \|\varepsilon(p_{\varepsilon}-p_{\mbox{\tinyP!P}})\|_{H^{1}(\Omega)}\leq c for all $\varepsilon>0$. It implies that there exists a subsequence of (\varepsilon(u_{\varepsilon}-u_{\mbox{\tinyP!P}}),\varepsilon(p_{\varepsilon}-p_{\mbox{\tinyP!P}})) which converges weakly to $(v^{(1)},q^{(1)})\in{H^{1}_{0}(\Omega)}^{n}\times Q$ if $\varepsilon\rightarrow\infty$. The next theorem states properties of the limit functions $v^{(1)}$ and $q^{(1)}$.

Theorem 3.4

Let v^{(1)}_{\varepsilon}:=\varepsilon(u_{\varepsilon}-u_{\mbox{\tinyP!P}})\in{H^{1}_{0}(\Omega)}^{n},q^{(1)}_{\varepsilon}:=\varepsilon(p_{\varepsilon}-p_{\mbox{\tinyP!P}})\in Q and let $(v^{(1)},q^{(1)})\in{H^{1}_{0}(\Omega)}^{n}\times Q$ satisfy

[TABLE]

Then there exists a constant $c>0$ independent of $\varepsilon$ such that

[TABLE]

Proof. The existence and the uniqueness of the pair $(v^{(1)},q^{(1)})\in{H^{1}_{0}(\Omega)}^{n}\times Q$ as a solution to (3.69) follows from Theorem 2.6. As in (3.66), we have

[TABLE]

Subtracting (3.69) from (3.72), it holds that

[TABLE]

Hence,

[TABLE]

where $v_{\varepsilon}:=v^{(1)}_{\varepsilon}-v^{(1)},q_{\varepsilon}:=q^{(1)}_{\varepsilon}-q^{(1)}$ and $h:=-{\operatorname{div}}v^{(1)}$. By Lemma 3.1 , we have

[TABLE]

for all $\varepsilon>0$. ∎

Next, we generalize Theorem 3.4 to the following theorem:

Theorem 3.5

Let $k\in{\mathbb{N}}$ be arbitrary ($k\geq 1$) and let v^{(0)}:=u_{\mbox{\tinyP!P}}. If functions $v^{(1)},v^{(2)},\cdots,v^{(k)}\in{H^{1}_{0}(\Omega)}^{n}$ and $q^{(1)},q^{(2)},\cdots,q^{(k)}\in Q$ satisfy

[TABLE]

for all $1\leq i\leq k$, then there exists a constant $c>0$ independent of $\varepsilon$ satisfying

[TABLE]

Proof. Let $(v^{(i)}_{\varepsilon},q^{(i)}_{\varepsilon})\in{H^{1}_{0}(\Omega)}^{n}\times Q~{}(1\leq i\leq k)$ satisfy

[TABLE]

Subtracting (3.75) from (3.78), it holds that

[TABLE]

Setting $v_{\varepsilon}:=v^{(i)}_{\varepsilon}-v^{(i)},q_{\varepsilon}:=q^{(i)}_{\varepsilon}-q^{(i)}$ and $h:=-{\operatorname{div}}v^{(i)}$, we obtain from Lemma 3.1 that the estimates

[TABLE]

hold for all $\varepsilon>0$. In particular, putting $i:=k$, we obtain

[TABLE]

[TABLE]

for all $\varepsilon>0$. By the uniqueness of the solution to (ES’) in Theorem 2.7, it leads that $v^{(i+1)}_{\varepsilon}=\varepsilon(v^{(i)}_{\varepsilon}-v^{(i)}),q^{(i+1)}_{\varepsilon}=\varepsilon(q^{(i)}_{\varepsilon}-q^{(i)})$ for all $i=1,\cdots,k-1$, and thus

[TABLE]

[TABLE]

Hence it holds that

[TABLE]

∎

Remark 3.6

Theorem 3.5 can be interpreted from the operator theory.

Let $t\geq 0,X:={H^{1}_{0}(\Omega)}^{n}\times Q,Y:={H^{-1}(\Omega)}^{n}\times Q^{*}$ be equipped with norms

[TABLE]

for $(u,p)\in X,(f,g)\in Y$, and let $A$ and $B$ be

[TABLE]

Then (u_{\mbox{\tinyP!P}},p_{\mbox{\tinyP!P}}) and $(u_{\varepsilon},p_{\varepsilon})$ satisfy

[TABLE]

where $f=(F,G)$. We have $A+tB\in{\rm Isom}(X,Y)$ for an arbitrary $t\geq 0$ by the analogy of Theorem 2.6 ($t=0$) and Theorem 2.7 ($t=1/\varepsilon$). Equation (3.75) states that

[TABLE]

for $i=1,\cdots,k$, i.e.

[TABLE]

By Theorem 3.5, there exists a constant $c>0$ such that

[TABLE]

for all $\varepsilon>0,f\in Y$.

4 Convergence of (ES) to (S)

In this section, we show that $(u_{\varepsilon},p_{\varepsilon})$ converges to (u_{\mbox{\tinyS}},p_{\mbox{\tinyS}}) weakly in ${H^{1}_{0}(\Omega)}^{n}\times({L^{2}(\Omega)}/{\mathbb{R}})$ as $\varepsilon\rightarrow 0$. Moreover, if p_{\mbox{\tinyS}}\in{H^{1}(\Omega)}, then $(u_{\varepsilon},p_{\varepsilon})$ converges to (u_{\mbox{\tinyS}},p_{\mbox{\tinyS}}) strongly in ${H^{1}_{0}(\Omega)}^{n}\times({L^{2}(\Omega)}/{\mathbb{R}})$ as $\varepsilon\rightarrow 0$.

The outline of the proof of our convergence results (Theorem 4.2, 4.3 and 4.4) is as follows. First, we prove the boundedness of the sequence $((u_{\varepsilon},p_{\varepsilon}))_{\varepsilon>0}$ in ${H^{1}_{0}(\Omega)}^{n}\times({L^{2}(\Omega)}/{\mathbb{R}})$. By the reflexivity of ${H^{1}_{0}(\Omega)}^{n}\times({L^{2}(\Omega)}/{\mathbb{R}})$, the sequence has a subsequence converging weakly in ${H^{1}_{0}(\Omega)}^{n}\times({L^{2}(\Omega)}/{\mathbb{R}})$. In the end, we show that the limit pair of functions satisfies (S’).

We start this section with a useful lemma.

Lemma 4.1

If $v\in{H^{1}(\Omega)}^{n},q\in{L^{2}(\Omega)}$ and $f\in{H^{-1}(\Omega)}^{n}$ satisfy

[TABLE]

then there exists a constant $c>0$ such that

[TABLE]

Proof. Let $c$ be the constant from Theorem 2.2. Then we obtain

[TABLE]

∎

Theorem 4.2

There exists a constant $c>0$ independent of $\varepsilon$ such that

[TABLE]

Furthermore, if $C^{\infty}_{0}(\Omega)\subset Q$, then we obtain

[TABLE]

See Appendix A for the proof.

If we add a regularity assumption of p_{\mbox{\tinyS}}, then $(u_{\varepsilon},p_{\varepsilon})$ converges strongly in ${H^{1}(\Omega)}^{n}\times{L^{2}(\Omega)}/{\mathbb{R}}$

Theorem 4.3

Suppose that p_{\mbox{\tinyS}}\in{H^{1}(\Omega)}. Then we obtain

[TABLE]

See Appendix A for the proof.

Theorem 4.3 does not give the convergence rate. If $Q={H^{1}(\Omega)}/{\mathbb{R}}$ (corresponding to the Neumann boundary condition (1.16)), then the convergence rate becomes $\sqrt{\varepsilon}$.

Theorem 4.4

Suppose that $Q={H^{1}(\Omega)}/{\mathbb{R}}$ and p_{\mbox{\tinyS}}\in{H^{1}(\Omega)}. Then there exists a constant $c>0$ independent of $\varepsilon$ such that

[TABLE]

Proof. We obtain from (ES’) and (S’) that

[TABLE]

Putting \varphi:=u_{\varepsilon}-u_{\mbox{\tinyS}}\in{H^{1}_{0}(\Omega)}^{n} and \psi:=p_{\varepsilon}-p_{\mbox{\tinyS}}\in{H^{1}(\Omega)}/{\mathbb{R}}, we get

[TABLE]

Subtracting \varepsilon\int_{\Omega}\nabla p_{\mbox{\tinyS}}\cdot\nabla(p_{\varepsilon}-p_{\mbox{\tinyS}}) from both sides of (4.83), we obtain

[TABLE]

To clarify the following estimates, we set \alpha:=\|\nabla(u_{\varepsilon}-u_{\mbox{\tinyS}})\|_{{L^{2}(\Omega)}^{n\times n}},\beta:=\|\nabla(p_{\varepsilon}-p_{\mbox{\tinyS}})\|_{{L^{2}(\Omega)}^{n}},a:=\|\nabla p_{\mbox{\tinyS}}\|_{{L^{2}(\Omega)}^{n}}+\|G\|_{({H^{1}(\Omega)}/{\mathbb{R}})^{*}}. The estimate (4.87) reads as

[TABLE]

Hence, $\alpha\leq a\sqrt{\varepsilon}/2$, i.e., \|\nabla(u_{\varepsilon}-u_{\mbox{\tinyS}})\|_{{L^{2}(\Omega)}^{n\times n}}\leq(\sqrt{\varepsilon}/2)(\|\nabla p_{\mbox{\tinyS}}\|_{{L^{2}(\Omega)}^{n}}+\|G\|_{({H^{1}(\Omega)}/{\mathbb{R}})^{*}}). By Lemma 4.1, we obtain

[TABLE]

∎

5 Numerical examples

For our simulations, we consider $\Omega=(0,1)\times(0,1)$. We take the following boundary conditions:

[TABLE]

on $\Gamma$. The exact solutions for (PP1) are u_{\mbox{\tinyP!P}}=(x(x-1),y(y-1))^{T} and p_{\mbox{\tinyP!P}}=2x+2y-2. We solve the problems (PP1), (ES1) and (S’) numerically by using the finite element method with P2/P1 elements by the software FreeFem++ FreeFem . The numerical solutions (u_{\mbox{\tinyP!P}},p_{\mbox{\tinyP!P}}),(u_{\varepsilon},p_{\varepsilon})~{}(\varepsilon=1,10^{-2}{\rm~{}or~{}}10^{-4}) and (u_{\mbox{\tinyS}},p_{\mbox{\tinyS}}) to the problems (PP1), (ES1) and (S’), respectively, are illustrated in Fig. 2–4. From these pictures we observe that $(u_{\varepsilon},p_{\varepsilon})$ seems to converge to (u_{\mbox{\tinyP!P}},p_{\mbox{\tinyP!P}}) as $\varepsilon\rightarrow\infty$ and to (u_{\mbox{\tinyS}},p_{\mbox{\tinyS}}) as $\varepsilon\rightarrow 0$ (as expected from Theorem 3.2 and 4.3.)

Next we compute the error estimate between the numerical solutions of (ES1) and (PP1). The numerical errors \|u_{\varepsilon}-u_{\mbox{\tinyP!P}}\|_{{L^{2}(\Omega)}^{n}}, \|\nabla(u_{\varepsilon}-u_{\mbox{\tinyP!P}})\|_{{L^{2}(\Omega)}^{n\times n}}, \|p_{\varepsilon}-p_{\mbox{\tinyP!P}}\|_{{L^{2}(\Omega)}} and \|\nabla(p_{\varepsilon}-p_{\mbox{\tinyP!P}})\|_{{L^{2}(\Omega)}^{n}} are shown in Fig. 5 and Fig. 6. Based on these values, we have fitted a constant $c$ such that \|u_{\varepsilon}-u_{\mbox{\tinyP!P}}\|_{{H^{1}(\Omega)}^{n}}\sim c/\varepsilon and \|p_{\varepsilon}-p_{\mbox{\tinyP!P}}\|_{{H^{1}(\Omega)}}\sim c/\varepsilon for $\varepsilon$ large. Fig. 5 and Fig. 6 indicate that there exists a constant $c$ such that \|u_{\varepsilon}-u_{\mbox{\tinyP!P}}\|_{{H^{1}(\Omega)}^{n}}\leq c/\varepsilon and \|p_{\varepsilon}-p_{\mbox{\tinyP!P}}\|_{{H^{1}(\Omega)}}\leq c/\varepsilon, as expected from Theorem 3.2.

We also compute the error estimate between the problems (ES1) and (S’) by numerical calculation. The numerical error estimate \|u_{\varepsilon}-u_{\mbox{\tinyS}}\|_{{L^{2}(\Omega)}^{n}},~{}\|\nabla(u_{\varepsilon}-u_{\mbox{\tinyS}})\|_{{L^{2}(\Omega)}^{n\times n}}~{},\|p_{\varepsilon}-p_{\mbox{\tinyS}}\|_{{L^{2}(\Omega)}} and \|\nabla(p_{\varepsilon}-p_{\mbox{\tinyS}})\|_{{L^{2}(\Omega)}^{n}} are shown in Fig. 7 and Fig. 8. Based on these values, we have fitted a constant $c$ such that \|u_{\varepsilon}-u_{\mbox{\tinyS}}\|_{{H^{1}(\Omega)}^{n}}\sim c\varepsilon and \|p_{\varepsilon}-p_{\mbox{\tinyS}}\|_{{L^{2}(\Omega)}}\sim c\varepsilon for $\varepsilon$ small. Fig. 7 and Fig. 8 indicate that there exists a constant $\tilde{c}$ such that \|u_{\varepsilon}-u_{\mbox{\tinyS}}\|_{{H^{1}(\Omega)}^{n}}\leq\tilde{c}\sqrt{\varepsilon} and \|p_{\varepsilon}-p_{\mbox{\tinyS}}\|_{{L^{2}(\Omega)}}\leq\tilde{c}\sqrt{\varepsilon}, as expected from Theorem 4.4.

Appendix

Theorem 2.7, 4.2, 4.3, and Proposition 2.8 are generalizations of several theorems stated in prev . 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 $u_{1}\in{H^{1}(\Omega)}^{n}$ with $\gamma_{0}u_{1}=u_{b}$. Since ${\operatorname{div}}:{H^{1}_{0}(\Omega)}^{n}\rightarrow{L^{2}(\Omega)}/{\mathbb{R}}$ is surjective (Girault, , Corollary 2.4, 2∘), there exists $u_{2}\in{H^{1}_{0}(\Omega)}^{n}$ such that ${\operatorname{div}}u_{2}={\operatorname{div}}u_{1}$. We put

[TABLE]

and note that $\gamma_{0}u_{0}=u_{b}$ and ${\operatorname{div}}u_{0}=0$. To simplify the notation, we set $u:=u_{\varepsilon}-u_{0}\in{H^{1}_{0}(\Omega)}^{n},p:=p_{\varepsilon}-p_{b}\in Q$, and define $f\in{H^{-1}(\Omega)}^{n}$ and $g\in Q^{*}$ by

[TABLE]

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

[TABLE]

Adding the equations in (.7), we get

[TABLE]

We check that $(\cdot,\cdot)_{\varepsilon}$ is a continuous coercive bilinear form on ${H^{1}_{0}(\Omega)}^{n}\times Q$. The bilinearity and continuity of $(\cdot,\cdot)_{\varepsilon}$ are obvious. The coercivity of $(\cdot,\cdot)_{\varepsilon}$ is obtained in the following way. Take $(v,q)^{T}\in{H^{1}_{0}(\Omega)}^{n}\times Q$. We have the following sequence of inequalities:

[TABLE]

Summarizing, $(\cdot,\cdot)_{\varepsilon}$ is a continuous coercive bilinear form and ${H^{1}_{0}(\Omega)}^{n}\times Q$ is a Hilbert space. Therefore, the conclusion of Theorem 2.7 follows from the Lax–Milgram Theorem. ∎

Let (u_{\mbox{\tinyS}},p_{\mbox{\tinyS}}),(u_{\mbox{\tinyP!P}},p_{\mbox{\tinyP!P}}) and $(u_{\varepsilon},p_{\varepsilon})$ be the solutions of (S’), (PP’) and (ES’), respectively, as guaranteed by Theorem 2.5, 2.6 and 2.7. We show that the subtract p_{\mbox{\tinyS}}-p_{\mbox{\tinyP!P}} satisfies

[TABLE]

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

Proof of Proposition 2.8. First, we prove that there exists a constant $c>0$ independent of $\varepsilon$ such that \|u_{\mbox{\tinyS}}-u_{\mbox{\tinyP!P}}\|_{{H^{1}(\Omega)}^{n}}\leq c\|\gamma_{0}p_{\mbox{\tinyS}}-\gamma_{0}p_{\mbox{\tinyP!P}}\|_{H^{1/2}(\Gamma)}, and if \gamma_{0}(p_{\mbox{\tinyS}}-p_{\mbox{\tinyP!P}})=0, then p_{\mbox{\tinyP!P}}=p_{\mbox{\tinyS}}. Taking the divergence of the first equation of (S’), we obtain

[TABLE]

in distributions sense. Since p_{\mbox{\tinyS}}\in{H^{1}(\Omega)} and $C^{\infty}_{0}(\Omega)$ is dense in ${H^{1}_{0}(\Omega)}$, it follows that

[TABLE]

for all $\psi\in{H^{1}_{0}(\Omega)}$. Together with (S’), (PP’) and ${H^{1}_{0}(\Omega)}\subset Q$, we obtain

[TABLE]

from the assumption $\langle G,\psi\rangle=\int_{\Omega}\nabla F\cdot\psi$. Putting \varphi:=u_{\mbox{\tinyS}}-u_{\mbox{\tinyP!P}}\in{H^{1}_{0}(\Omega)}^{n} in (.10), we get

[TABLE]

Hence,

[TABLE]

holds. From the second equation of (.10), we obtain

[TABLE]

for all $\psi\in{H^{1}_{0}(\Omega)}$. Thus we find

[TABLE]

Since $\gamma_{0}$ is surjective and the space $\mbox{Ker}(\gamma_{0})={H^{1}_{0}(\Omega)}$, ${H^{1}(\Omega)}/{H^{1}_{0}(\Omega)}$ and $H^{1/2}(\Gamma)$ are isomorphic, there exists a constant $c_{2}>0$ such that $\|q\|_{{H^{1}(\Omega)}/{H^{1}_{0}(\Omega)}}\leq c_{2}\|\gamma_{0}q\|_{H^{1/2}(\Gamma)}$ for all $q\in{H^{1}(\Omega)}$. Hence, we obtain

[TABLE]

Together with (.11), we obtain \|u_{\mbox{\tinyS}}-u_{\mbox{\tinyP!P}}\|_{{H^{1}(\Omega)}^{n}}\leq c_{1}c_{2}\|\gamma_{0}p_{\mbox{\tinyS}}-\gamma_{0}p_{\mbox{\tinyP!P}}\|_{H^{1/2}(\Gamma)}. Moreover, if \gamma_{0}(p_{\mbox{\tinyS}}-p_{\mbox{\tinyP!P}})=0, then p_{\mbox{\tinyP!P}}=p_{\mbox{\tinyS}}.

Next, we prove that there exists a constant $c>0$ independent of $\varepsilon$ such that \|u_{\mbox{\tinyS}}-u_{\varepsilon}\|_{{H^{1}(\Omega)}^{n}}\leq c\|\gamma_{0}p_{\mbox{\tinyS}}-\gamma_{0}p_{\varepsilon}\|_{H^{1/2}(\Gamma)}, and if \gamma_{0}(p_{\mbox{\tinyS}}-p_{\mbox{\tinyP!P}})=0, then p_{\mbox{\tinyP!P}}=p_{\varepsilon}. Let w_{\varepsilon}:=u_{\mbox{\tinyS}}-u_{\varepsilon}\in{H^{1}_{0}(\Omega)}^{n} and r_{\varepsilon}:=p_{\mbox{\tinyP!P}}-p_{\varepsilon}\in Q. By (S’), (PP’) and (ES’), we obtain

[TABLE]

Putting $\varphi:=w_{\varepsilon}$ and $\psi:=r_{\varepsilon}$ and adding the two equations of (.19), we get

[TABLE]

from $\int_{\Omega}(\nabla r_{\varepsilon})\cdot w_{\varepsilon}=-\int_{\Omega}({\operatorname{div}}w_{\varepsilon})r_{\varepsilon}$. Thus we find

[TABLE]

Together with (.16), we obtain

[TABLE]

Moreover, by (.20), we obtain

[TABLE]

Hence, if \gamma_{0}(p_{\mbox{\tinyS}}-p_{\mbox{\tinyP!P}})=0, then p_{\mbox{\tinyP!P}}=p_{\varepsilon}. ∎

We show that the sequence $((u_{\varepsilon},p_{\varepsilon}))_{\varepsilon>0}$ is bounded in ${H^{1}(\Omega)}^{n}\times({L^{2}(\Omega)}/{\mathbb{R}})$. By the reflexivity of ${H^{1}(\Omega)}^{n}\times({L^{2}(\Omega)}/{\mathbb{R}})$, the sequence $((u_{\varepsilon},p_{\varepsilon}))_{\varepsilon>0}$ has a subsequence converging weakly to somewhere in ${H^{1}(\Omega)}^{n}\times({L^{2}(\Omega)}/{\mathbb{R}})$. 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 $u_{b}\in{H^{1}(\Omega)}^{n},f\in{H^{-1}(\Omega)}^{n}$ and $g\in Q^{*}$ as (.1) and (.4) in the proof of Theorem 2.7. We put $\tilde{u}_{\varepsilon}:=u_{\varepsilon}-u_{b}\in{H^{1}_{0}(\Omega)}^{n},\tilde{p}_{\varepsilon}:=p_{\varepsilon}-p_{b}\in Q$. Then we obtain

[TABLE]

Putting $\varphi:=\tilde{u}_{\varepsilon},\psi:=\tilde{p}_{\varepsilon}$ and adding the two equations of (.23), we get

[TABLE]

since $\int_{\Omega}(\nabla\tilde{p}_{\varepsilon})\cdot\tilde{u}_{\varepsilon}=-\int_{\Omega}({\operatorname{div}}\tilde{u}_{\varepsilon})\tilde{p}_{\varepsilon}$. Hence, $(\|\tilde{u}_{\varepsilon}\|_{{H^{1}(\Omega)}^{n}})_{0<\varepsilon<1}$ and $(\|\sqrt{\varepsilon}\tilde{p}_{\varepsilon}\|_{H^{1}(\Omega)})_{0<\varepsilon<1}$ are bounded. Moreover, by Lemma 4.1, we obtain

[TABLE]

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

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

[TABLE]

Hence, from (.23) with $\varepsilon:=\varepsilon_{k}$, taking $k\rightarrow\infty$, we obtain

[TABLE]

where we have used that

[TABLE]

[TABLE]

as $k\rightarrow\infty$. By (.4), the first equation of (.26) implies that

[TABLE]

for all $\varphi\in{H^{1}_{0}(\Omega)}^{n}$. From the second equation of (.26) and $C^{\infty}_{0}(\Omega)\subset Q$, ${\operatorname{div}}(u+u_{b})=0$ follows. Hence, we obtain that $(u+u_{b},p+[p_{b}])$ satisfies (S’), i.e., u_{\mbox{\tinyS}}=u+u_{b} and p_{\mbox{\tinyS}}=p+[p_{b}]. Then we have

[TABLE]

[TABLE]

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

Using Theorem 4.2 and the Rellich-Kondrachov Theorem, it is easy to prove Theorem 4.3.

*Proof of Theorem *4.3. We have from (ES’) and (S’) that

[TABLE]

Putting \varphi:=u_{\varepsilon}-u_{\mbox{\tinyS}}\in{H^{1}_{0}(\Omega)}^{n} and \tilde{p}_{\mbox{\tinyS}}:=p_{\mbox{\tinyS}}-p_{b}\in{H^{1}(\Omega)}, we get

[TABLE]

since -\int_{\Omega}(\nabla(p_{\varepsilon}-p_{b}))\cdot(u_{\varepsilon}-u_{\mbox{\tinyS}})=\int_{\Omega}(p_{\varepsilon}-p_{b}){\operatorname{div}}(u_{\varepsilon}-u_{\mbox{\tinyS}}) and {\operatorname{div}}u_{\mbox{\tinyS}}=0. Thus,

[TABLE]

Putting $\psi:=p_{\varepsilon}-p_{b}\in Q$, we have

[TABLE]

Hence,

[TABLE]

Together with (.27) and (.28), we obtain

[TABLE]

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

[TABLE]

Therefore,

[TABLE]

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

[TABLE]

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