Convergence of a finite volume scheme for the incompressible fluids

TL;DR

This paper proves the convergence of a finite volume scheme for 2D incompressible Navier-Stokes equations using a triangular mesh and a projection method, building on previous stability results.

Contribution

It establishes the convergence of a specific finite volume scheme for incompressible fluids, extending prior stability analysis.

Findings

Scheme is stable and convergent for 2D incompressible Navier-Stokes

Uses triangular mesh with piecewise constant velocity and affine pressure

Builds on previous stability proof to show convergence

Abstract

We consider a finite volume scheme for the two-dimensional incompressible Navier-Stokes equations. We use a triangular mesh. The unknowns for the velocity and pressure are respectively piecewise constant and affine. We use a projection method to deal with the incompressibility constraint. In a former paper, the stability of the scheme has been proven. We infer from it its convergence.

Equations

$${\mathbf{u}}_t-\frac{1}{Re}\mathbf{\Delta u}+(\mathbf{u}\cdot \text{boldmath}\nabla \text{unboldmath})\mathbf{u}+\nabla p=\mathbf{f},$$

$$div\mathbf{u}=0,$$

$$\mathbf{u}{|}_{\partial \Omega }=0,\mathbf{u}{|}_{t=0}={\mathbf{u}}_0.$$

$$\forall \sigma \in {\mathcal{E}}_h,d({\mathbf{x}}_{K_{\sigma }},\sigma )\ge C|\sigma |and|\sigma |\ge Ch.$$

$${\tau }_{\sigma }\ge C$$

$$|K|=\frac{1}{2}|\sigma |h_{K,\sigma }\ge \frac{1}{2}|\sigma |d({\mathbf{x}}_K,{\mathbf{x}}_{\sigma })\ge C{h}^2.$$

$$P_0=\{q\in L^2;\forall K\in {\mathcal{T}}_h,q{|}_{K}is\; a\; constant\},{\mathbf{P}}_0=(P_0{)}^2.$$

$$\Vert {\mathbf{v}}_h{\Vert }_h={\left(\sum _{\sigma \in {\mathcal{E}}_h^{int}}{\tau }_{\sigma }|{\mathbf{v}}_{L_{\sigma }}-{\mathbf{v}}_{K_{\sigma }}{|}^2+\sum _{\sigma \in {\mathcal{E}}_h^{ext}}{\tau }_{\sigma }|{\mathbf{v}}_{K_{\sigma }}{|}^2\right)}^{1/2}.$$

$$|{\mathbf{v}}_h|\le C\Vert {\mathbf{v}}_h{\Vert }_h.$$

$$\Vert {\mathbf{v}}_h{\Vert }_{-1,h}=\underset{{\text{boldmath}\psi \text{unboldmath}}_h\in {\mathbf{P}}_0}{\sup }\frac{({\mathbf{v}}_h,{\text{boldmath}\psi \text{unboldmath}}_h)}{\Vert {\text{boldmath}\psi \text{unboldmath}}_h{\Vert }_h}.$$

$$\forall K\in {\mathcal{T}}_h,({\Pi }_{{\mathbf{P}}_0}\mathbf{w}){|}_K=\frac{1}{|K|}{\int }_K\mathbf{w}(\mathbf{x})d\mathbf{x}.$$

$$\forall K\in {\mathcal{T}}_h,{\tilde{\Pi }}_{{\mathbf{P}}_0}\mathbf{v}{|}_K=\mathbf{v}({\mathbf{x}}_K).$$

$$|\mathbf{v}-{\tilde{\Pi }}_{{\mathbf{P}}_0}\mathbf{v}|\le Ch\Vert \mathbf{v}{\Vert }_2$$

$$P_1^d$$

$$P_1^{nc}$$

$${\nabla }_h{q}_h{|}_K=\frac{1}{|K|}{\int }_K\nabla q_{h}d\mathbf{x}.$$

$$\forall \sigma \in {\mathcal{E}}_h,{\int }_{\sigma }({\Pi }_{P_1^{nc}}q)d\sigma ={\int }_{\sigma }qd\sigma .$$

$$\left|\nabla q-{\nabla }_h({\Pi }_{P_1^{nc}}q)\right|\le Ch\Vert q{\Vert }_2,|\mathbf{v}-{\Pi }_{{\mathbf{P}}_1^{nc}}\mathbf{v}|\le Ch\Vert \mathbf{v}{\Vert }_1,$$

$$\mathbf{R}{\mathbf{T}}_0^{\mathbf{d}}$$

$$\mathbf{R}{\mathbf{T}}_0$$

$$\forall K\in {\mathcal{T}}_h,\forall \sigma \in {\mathcal{E}}_K,({\Pi }_{\mathbf{R}{\mathbf{T}}_0}\mathbf{v}\cdot {\mathbf{n}}_{K,\sigma }{)}_{\sigma }=\frac{1}{|\sigma |}{\int }_{\sigma }\mathbf{v}d\sigma .$$

$$|\mathbf{v}-{\Pi }_{\mathbf{R}{\mathbf{T}}_0}\mathbf{v}|\le Ch\Vert \mathbf{v}{\Vert }_1.$$

$$\forall \sigma \in {\mathcal{E}}_h^{int},({div}_h{\mathbf{v}}_h)({\mathbf{x}}_{\sigma })=\frac{3|\sigma |}{|K_{\sigma }|+|L_{\sigma }|}({\mathbf{v}}_{L_{\sigma }}-{\mathbf{v}}_{K_{\sigma }})\cdot {\mathbf{n}}_{K,\sigma };$$

$$\forall \sigma \in {\mathcal{E}}_h^{ext},({div}_h{\mathbf{v}}_h)({\mathbf{x}}_{\sigma })=-\frac{3|\sigma |}{|K_{\sigma }|+|L_{\sigma }|}{\mathbf{v}}_{K_{\sigma }}\cdot {\mathbf{n}}_{K,\sigma }.$$

$$\forall q_h\in P_1^{nc},{\Delta }_h{q}_h={div}_h({\nabla }_h{q}_h).$$

$${\tilde{\mathbf{\Delta }}}_h{\mathbf{v}}_h{|}_K=\frac{1}{|K|}\sum _{\sigma \in {\mathcal{E}}_K\cap {\mathcal{E}}_h^{int}}{\tau }_{\sigma }({\mathbf{v}}_{L_{\sigma }}-{\mathbf{v}}_{K_{\sigma }})-\frac{1}{|K|}\sum _{\sigma \in {\mathcal{E}}_K\cap {\mathcal{E}}_h^{ext}}{\tau }_{\sigma }{\mathbf{v}}_{K_{\sigma }}.$$

$${\widetilde{\hbox{\rm\bf b}}}_{h}({\bf u}_{h},{\bf v}_{h})\big{|}_{K}=\frac{1}{|K|}\sum_{\sigma\in{\mathcal{E}}_{K}\cap{\mathcal{E}}^{int}_{h}}\,|\sigma|\,\Big{(}({\bf u}\cdot{\mathbf{n}}_{K,\sigma})^{+}_{\sigma}\,{\bf v}_{K}+({\bf u}\cdot{\mathbf{n}}_{K,\sigma})^{-}_{\sigma}\,{\bf v}_{L_{\sigma}}\Big{)}.$$

$${\hbox{\rm b}}_{h}({\bf u}_{h},{\bf v}_{h},{\bf w}_{h})=\sum_{K\in{\mathcal{T}}_{h}}|K|\,{\bf w}_{K}\cdot{\widetilde{\hbox{\rm\bf b}}}_{h}({\bf u}_{h},{\bf v}_{h})\big{|}_{K}.$$

$${\mathbf{u}}_h^0\in {\mathbf{P}}_0\cap \mathbf{R}{\mathbf{T}}_0,{\mathbf{u}}_h^1\in {\mathbf{P}}_0\cap \mathbf{R}{\mathbf{T}}_0,p_h^1\in P_1^{nc}\cap L_0^2.$$

$$\frac{3{\tilde{\mathbf{u}}}_h^{n+1}-4{\mathbf{u}}_h^n+{\mathbf{u}}_h^{n-1}}{2k}$$

$$+{\tilde{\mathbf{b}}}_h(2{\mathbf{u}}_h^n-{\mathbf{u}}_h^{n-1},{\tilde{\mathbf{u}}}_h^{n+1})+{\nabla }_h{p}_h^n={\mathbf{f}}_h^{n+1},$$

$${\Delta }_h(p_h^{n+1}-p_h^n)=\frac{3}{2k}{div}_h{\tilde{\mathbf{u}}}_h^{n+1},$$

$${\mathbf{u}}_h^{n+1}={\tilde{\mathbf{u}}}_h^{n+1}-\frac{2k}{3}{\nabla }_h(p_h^{n+1}-p_h^n).$$

Full text

††journal: Numerische Mathematik

1 Introduction

We consider the flow of an incompressible fluid in a open bounded polyhedral set $\Omega\subset\mathbb{R}^{2}$ during the time interval $[0,T]$. The velocity field $\mathbf{u}:\Omega\times[0,T]\to\mathbb{R}^{2}$ and the pressure field $p:\Omega\times[0,T]\to\mathbb{R}$ satisfy the Navier-Stokes equations

[TABLE]

with the boundary and initial condition

[TABLE]

The terms $\mathbf{\Delta}{\bf u}$ and ({\bf u}\cdot{\hbox{\boldmath\nabla \unboldmath\!\!}}){\bf u} are associated with the physical phenomena of diffusion and convection, respectively. The Reynolds number Re measures the influence of convection in the flow. For equations (1)–(2), finite element and finite difference methods are well known and mathematical studies are available (see giraultr for example). For finite volume schemes, numerous computations have been conducted (kimchoi and boicaya for example). However, few mathematical results are available in this case. Let us cite Eymard and Herbin herb3 and Eymard, Latché and Herbin eymard . In order to deal with the incompressibility constraint, these works use a penalization method. Another way is to use the projection methods which have been introduced by Chorin chorin and Temam temam . This is the case in Faure faure where the mesh is made of squares. In Zimmermann zimm1 the mesh is made of triangles, which allows more complex geometries. In the present paper the mesh is also made of triangles, but we consider a different discretisation for the pressure. It leads to a linear system with a better-conditioned matrix. The layout of the article is the following. We first introduce (section 2.1) some notations and hypotheses on the mesh. We define (section 2.2) the spaces we use to approximate the velocity and pressure. We define also (section 2.3) the operators we use to approximate the differential operators in (1)–(2). By combining this with a projection method, we build the scheme in section 3. In order to provide a mathematical analysis, we state in section 4 that the differential operators in (1)–(2) and their discrete counterparts share similar properties. In particular, the discrete operators for the gradient and the divergence are adjoint. We then prove in section 5 the convergence of the scheme.

We conclude with some notations. We denote by $\chi_{I}$ the characteristic function of an interval $I\subset\mathbb{R}$. We denote by ${\mathcal{C}}^{\infty}_{0}={\mathcal{C}}^{\infty}_{0}(\Omega)$ the set of the functions with a compact support in $\Omega$. The spaces $(L^{2},|.|)$ and $(L^{\infty},\|.\|_{\infty})$ are the usual Lebesgue spaces and we set $L^{2}_{0}=\{q\in L^{2}\,;\int_{\Omega}q\,d{\bf x}=0\}$. Their vectorial counterparts are $({\bf L}^{2},|.|)$ and $({\bf L}^{\infty},\|.\|_{\infty})$ with ${\bf L}^{2}=(L^{2})^{2}$ and ${\bf L}^{\infty}=(L^{\infty})$. For $k\in\mathbb{N}^{*}$, $(H^{k},\|\cdot\|_{k})$ is the usual Sobolev space. Its vectorial counterpart is $({\bf H}^{k},\|.\|_{k})$ with ${\bf H}^{k}=(H^{k})^{2}$. For $k=1$, the functions of ${\bf H}^{1}$ with a null trace on the boundary form the space ${\bf H}^{1}_{0}$. Also, we set {\hbox{\boldmath\nabla \unboldmath\!\!}}{\bf u}=(\nabla u_{1},\nabla u_{2})^{T} if ${\bf u}=(u_{1},u_{2})\in{\bf H}^{1}$. If $\mathbf{X}\subset{\bf L}^{2}$ is a Banach space, we define ${\mathcal{C}}(0,T;\mathbf{X})$ (resp. $L^{2}(0,T;\mathbf{X})$) as the set of the applications $\mathbf{g}:[0,T]\to\mathbf{X}$ such that $t\to|\mathbf{g}(t)|$ is continous (resp. square integrable). The norm $\|.\|_{{\mathcal{C}}(0,T;\mathbf{X})}$ is defined by $\|\mathbf{g}\|_{{\mathcal{C}}(0,T;\mathbf{X})}=\sup_{s\in[0,T]}|\mathbf{g}(s)|$. Finally in all calculations, $C$ is a generic positive constant, depending only on $\Omega$, ${\bf u}_{0}$ and ${\bf f}$.

2 Discrete setting

First, we introduce the spaces and operators needed to build the mesh.

2.1 The mesh

Let ${\mathcal{T}}_{h}$ be a triangular mesh of $\Omega$: $\overline{\Omega}=\cup_{K\in{\mathcal{T}}_{h}}K$. For each triangle $K\in{\mathcal{T}}_{h}$, we denote by $|K|$ its area and ${\mathcal{E}}_{K}$ the set of his edges. If $\sigma\in{\mathcal{E}}_{K}$, ${\mathbf{n}}_{K,\sigma}$ is the unit vector normal to $\sigma$ pointing outwards of $K$.

The set of edges of the mesh is ${\mathcal{E}}_{h}=\cup_{K\in{\mathcal{T}}_{h}}{\mathcal{E}}_{K}$. The length of an edge $\sigma\in{\mathcal{E}}_{h}$ is $|\sigma|$. The set of edges inside $\Omega$ (resp. on the boundary) is ${\mathcal{E}}^{int}_{h}$ (resp. ${\mathcal{E}}^{ext}_{h}$): ${\mathcal{E}}_{h}={\mathcal{E}}^{int}_{h}\cup{\mathcal{E}}^{ext}_{h}$. If $\sigma\in{\mathcal{E}}^{int}_{h}$, $K_{\sigma}$ and $L_{\sigma}$ are the triangles sharing $\sigma$ as an edge. If $\sigma\in{\mathcal{E}}^{ext}_{h}$, only the triangle $K_{\sigma}$ inside $\Omega$ is defined.

We denote by ${\bf x}_{K}$ the circumcenter of a triangle $K$. We assume that the measure of all interior angles of the triangles of the mesh are below $\frac{\pi}{2}$, so that ${\bf x}_{K}\in K$. If $\sigma\in{\mathcal{E}}^{int}_{h}$ (resp. $\sigma\in{\mathcal{E}}^{ext}_{h}$ ) we set $d_{\sigma}=d({\bf x}_{K_{\sigma}},{\bf x}_{L_{\sigma}})$ (resp. $d_{\sigma}=d({\bf x}_{\sigma},{\bf x}_{K_{\sigma}})$). We define for all edge $\sigma\in{\mathcal{E}}_{h}$: $\tau_{\sigma}=\frac{|\sigma|}{d_{\sigma}}$. The maximum circumradius of the triangles of the mesh is $h$. We assume that there exists $C>0$ such that

[TABLE]

It implies that there exists a constant $C>0$ such that for all edge $\sigma\in{\mathcal{E}}_{h}$

[TABLE]

and for all triangles $K\in{\mathcal{T}}_{h}$ we have (with $\sigma\in{\mathcal{E}}_{K}$ and $h_{K,\sigma}$ the matching altitude)

[TABLE]

2.2 The discrete spaces

We first define

[TABLE]

For the sake of concision, we set for all $q_{h}\in P_{0}$ (resp. ${\bf v}_{h}\in{\bf P}_{0}$) and all triangle $K\in{\mathcal{T}}_{h}$: $q_{K}=q_{h}|_{K}$ (resp. ${\bf v}_{K}={\bf v}_{h}|_{K}$). Although ${\bf P}_{0}\not\subset{\bf H}^{1}$, we define the discrete equivalent of a ${\bf H}^{1}$ norm as follow. For all ${\bf v}_{h}\in{\bf P}_{0}$ we set

[TABLE]

We have eymgal a discrete Poincaré inequality for ${\bf P}_{0}$: there exists $C>0$ such that for all ${\bf v}_{h}\in{\bf P}_{0}$

[TABLE]

From the norm $\|.\|_{h}$ we deduce a dual norm. For all ${\bf v}_{h}\in{\bf P}_{0}$ we set

[TABLE]

For all ${\bf u}_{h}\in{\bf P}_{0}$ and ${\bf v}_{h}\in{\bf P}_{0}$ we have $({\bf u}_{h},{\bf v}_{h})\leq\|{\bf u}_{h}\|_{-1,h}\,\|{\bf v}_{h}\|_{h}$. We define the projection operator $\Pi_{{\bf P}_{0}}:{\bf L}^{2}\to{\bf P}_{0}$ as follows. For all ${\bf w}\in{\bf L}^{2}$, $\Pi_{{\bf P}_{0}}{\bf w}\in{\bf P}_{0}$ is given by

[TABLE]

We easily check that for all ${\bf w}\in{\bf L}^{2}$ and ${\bf v}_{h}\in{\bf P}_{0}$ we have $(\Pi_{{\bf P}_{0}}{\bf w},{\bf v}_{h})=({\bf w},{\bf v}_{h})$. We deduce from this that $\Pi_{{\bf P}_{0}}$ is stable for the ${\bf L}^{2}$ norm. We define also the operator $\widetilde{\Pi}_{{\bf P}_{0}}:{\bf H}^{2}\to{\bf P}_{0}$. For all ${\bf v}\in{\bf H}^{2}$, $\widetilde{\Pi}_{{\bf P}_{0}}{\bf v}\in{\bf P}_{0}$ is given by

[TABLE]

According to the Sobolev embedding theorem, ${\bf v}\in{\bf H}^{2}$ is a.e. equal to a continuous function. Therefore the definition above makes sense. One checks zimm1 that there exists $C>0$ such that

[TABLE]

for all ${\bf v}\in{\bf H}^{1}$. We introduce also the finite element spaces

[TABLE]

If $q_{h}\in P^{nc}_{1}$, we have usually $\nabla q_{h}\not\in{\bf L}^{2}$. Therefore we define the operator $\nabla_{h}:P^{nc}_{1}\to{\bf P}_{0}$ by setting for all $q_{h}\in P_{0}$ and all triangle $K\in{\mathcal{T}}_{h}$

[TABLE]

We define the projection operator $\Pi_{P^{nc}_{1}}$. For all $q\in H^{1}$, $\Pi_{P^{nc}_{1}}q$ is given by

[TABLE]

We also set $\Pi_{{\bf P}^{nc}_{1}}=(\Pi_{P^{nc}_{1}})^{2}$. One checks that there exists $C>0$ such that

[TABLE]

for all $q\in H^{1}$ and ${\bf v}\in{\bf H}^{1}$.

We also use the Raviart-Thomas spaces

[TABLE]

For all ${\bf v}_{h}\in\mathbf{RT_{0}}$, $K\in{\mathcal{T}}_{h}$ and $\sigma\in{\mathcal{E}}_{K}$ we set $({\bf v}_{h}\cdot{\mathbf{n}}_{K,\sigma})_{\sigma}={\bf v}_{h}|_{K}\cdot{\mathbf{n}}_{K,\sigma}$. We define the operator $\Pi_{\mathbf{RT_{0}}}:{\bf H}^{1}\to\mathbf{RT_{0}}$. For all ${\bf v}\in{\bf H}^{1}$, $\Pi_{\mathbf{RT_{0}}}{\bf v}\in\mathbf{RT_{0}}$ is given by

[TABLE]

One checks brezzfor that there exists a constant $C>0$ such that for all ${\bf v}\in{\bf H}^{1}$

[TABLE]

2.3 The discrete operators

The equations (1)–(2) use the differential operators gradient, divergence and laplacian. Using the spaces of section 2.2, we now define their discrete counterparts. The discrete gradient $\nabla_{h}:P^{nc}_{1}\to{\bf P}_{0}$ is defined by (10). The discrete divergence operator $\hbox{div}_{h}:{\bf P}_{0}\to P^{nc}_{1}$ is built so that it is adjoint to the operator $\nabla_{h}$ (proposition 3 below). We set for all ${\bf v}_{h}\in{\bf P}_{0}$ and all triangle $K\in{\mathcal{T}}_{h}$

[TABLE]

The first discrete laplacian $\Delta_{h}:P^{nc}_{1}\to P^{nc}_{1}$ is given by

[TABLE]

The second discrete laplacian ${\mathbf{\widetilde{\Delta}}}_{h}:{\bf P}_{0}\to{\bf P}_{0}$ is the usual operator in finite volume schemes eymgal . We set for all ${\bf v}_{h}\in{\bf P}_{0}$ and all triangle $K\in{\mathcal{T}}_{h}$

[TABLE]

In order to approximate the convection term ({\bf u}\cdot{\hbox{\boldmath\nabla \unboldmath\!\!}}){\bf u} in (1), we define a bilinear form ${\widetilde{\hbox{\rm\bf b}}}_{h}:\mathbf{RT_{0}}\times{\bf P}_{0}\to{\bf P}_{0}$ using the well-known eymgal upwind scheme. For all ${\bf u}_{h}\in{\bf P}_{0}$, ${\bf v}_{h}\in{\bf P}_{0}$, and all triangle $K\in{\mathcal{T}}_{h}$ we set

[TABLE]

We have set $a^{+}=\max(a,0)$, $a^{-}=\min(a,0)$ for all $a\in\mathbb{R}$. Lastly, we define the trilinear form ${\hbox{\rm b}}_{h}:\mathbf{RT_{0}}\times{\bf P}_{0}\times{\bf P}_{0}\to\mathbb{R}$ as follows. For all ${\bf u}_{h}\in\mathbf{RT_{0}}$, ${\bf v}_{h}\in{\bf P}_{0}$, ${\bf w}_{h}\in{\bf P}_{0}$, we set

[TABLE]

3 The scheme

We have defined in section 2 the discretization in space. We now have to define the discretization in time, and treat the incompressibility constraint (2). We use a projection method to this end. This kind of method has been introduced by Chorin chorin and Temam temam . The time interval $[0,T]$ is split with a time step $k$: $[0,T]=\bigcup_{n=0}^{N}[t_{n},t_{n+1}]$ with $N\in\mathbb{N}^{*}$ et $t_{n}=n\,k$ for all $n\in\{0,\dots,N\}$. We start with the initial values

[TABLE]

For all $n\in\{1,\dots,N\}$, $({\tilde{\mathbf{u}}}^{n+1}_{h},p^{n+1}_{h},{\bf u}^{n+1}_{h})$ is deduced from $({\tilde{\mathbf{u}}}^{n}_{h},p^{n}_{h},{\bf u}^{n}_{h})$ as follows.

• •

${\tilde{\mathbf{u}}}^{n+1}_{h}\in{\bf P}_{0}$ is given by

[TABLE]

• •

$p^{n+1}_{h}\in P^{nc}_{1}\cap L^{2}_{0}$ is the solution of

[TABLE]

• •

${\bf u}^{n+1}_{h}\in{\bf P}_{0}$ is given by

[TABLE]

We have proven in zimm1 that the scheme is well defined. In particular the term ${\widetilde{\hbox{\rm\bf b}}}_{h}(2\,{\bf u}^{n}_{h}-{\bf u}^{n-1}_{h})$ in (• ‣ 3) is defined thanks to the following result.

Proposition 1

For $m\in\{0,\dots,N\}$ we have ${\bf u}^{m}_{h}\in\mathbf{RT_{0}}$.

Note also that for $m\in\{0,\dots,N\}$ we have $\hbox{div}\,{\bf u}^{m}_{h}=0$, since ${\bf u}^{m}_{h}\in{\bf P}_{0}$. Thus the incompressibility condition (2) is fullfilled.

4 Properties of the discrete operators

The operators defined in section 2.3 have the following properties zimm1 .

Proposition 2

There exists a constant $C>0$ such that for all ${\bf u}_{h}\in\mathbf{RT_{0}}$ satisfying $\hbox{ \rm div}\,{\bf u}_{h}=0$, ${\bf v}_{h}\in{\bf P}_{0}$, ${\bf w}_{h}\in{\bf P}_{0}$:

[TABLE]

Proposition 3

For all ${\bf v}_{h}\in{\bf P}_{0}$ and $q_{h}\in P^{nc}_{1}$: $({\bf v}_{h},\nabla_{h}q_{h})=-(q_{h},\hbox{\rm div}_{h}\,{\bf v}_{h})$.

Proposition 4

For all ${\bf u}_{h}\in{\bf P}_{0}$ and ${\bf v}_{h}\in{\bf P}_{0}$: $-({\mathbf{\widetilde{\Delta}}}_{h}{\bf u}_{h},{\bf u}_{h})=\|{\bf u}_{h}\|^{2}_{h}$ and $-({\mathbf{\widetilde{\Delta}}}_{h}{\bf u}_{h},{\bf v}_{h})\leq\|{\bf u}_{h}\|_{h}\,\|{\bf v}_{h}\|_{h}$.

If ${\bf v}\in{\bf H}^{1}$ we have $|\hbox{div}\,{\bf v}|\leq\|{\bf v}\|_{1}$. The operator $\hbox{div}_{h}$ has a similar property.

Proposition 5

There exists a constant $C>0$ such that for all ${\bf v}_{h}\in{\bf P}_{0}$

[TABLE]

Proof. Using a quadrature formula we have

[TABLE]

Let $K\in{\cal{T}}_{h}$. Using definition and (4) we have

[TABLE]

Thus: $|\hbox{div}_{h}\,{\bf v}_{h}|^{2}\leq C\sum_{K\in{\cal{T}}_{h}}\!\!\!\left(\sum_{\sigma\in{\cal{E}}_{K}\cap{\cal{E}}^{int}_{h}}|{\bf v}_{L_{\sigma}}-{\bf v}_{K}|^{2}+\sum_{\sigma\in{\cal{E}}_{K}\cap{\cal{E}}^{ext}_{h}}\!|{\bf v}_{K}|^{2}\right)$. Writing the sum over the triangles as a sum over the edges, we get

[TABLE]

Proposition 6

If ${\bf u}_{h}\in{\bf P}_{0}$ and ${\bf v}_{h}\in{\bf P}_{0}$ we have $({\mathbf{\widetilde{\Delta}}}_{h}{\bf u}_{h},{\bf v}_{h})=({\bf u}_{h},{\mathbf{\widetilde{\Delta}}}_{h}{\bf v}_{h})$.

Proof. Using definition (2.3) one checks that

[TABLE]

Proposition 7

There exists $C>0$ such that for all ${\bf v}\in{\bf H}^{2}$ satisfying {\hbox{\boldmath\nabla \unboldmath\!\!}}{\bf v}\cdot{\mathbf{n}}|_{\partial\Omega}=\mathbf{0}

[TABLE]

Proof. Let {\hbox{\boldmath\psi \unboldmath\!\!}}_{h}\in{\bf P}_{0}. We have

[TABLE]

For all $K\in{\cal{T}}_{h}$, using (2.3) and the divergence formula, we get

[TABLE]

Thus, by writing the sum over the triangles as a sum over the edges, we get

[TABLE]

with {\mathbf{R}}_{\sigma}=\int_{\sigma}\Big{(}{\hbox{\boldmath\nabla \unboldmath\!\!}}{\bf v}\cdot{\mathbf{n}}_{K_{\sigma},\sigma}-\frac{1}{d_{\sigma}}\,\big{(}{\bf v}({\bf x}_{L_{\sigma}})-{\bf v}({\bf x}_{K_{\sigma}})\big{)}\Big{)}\,d\sigma. We denote by $D_{\sigma}$ the quadrilatere defined by ${\bf x}_{K_{\sigma}}$, ${\bf x}_{L_{\sigma}}$ and the endpoints of $\sigma$. Using a Taylor expansion and a density argument, we get as in eymgal that

[TABLE]

Thus, using the Cauchy-Schwarz inequality, we get

[TABLE]

According to (3)

[TABLE]

Thus \left|\big{(}\Pi_{{\bf P}_{0}}(\mathbf{\Delta}{\bf v})-\mathbf{\Delta}_{h}(\widetilde{\Pi}_{{\bf P}_{0}}{\bf v}),{\hbox{\boldmath\psi \unboldmath\!\!}}_{h}\big{)}\right|\leq C\,h\,\|{\hbox{\boldmath\psi \unboldmath\!\!}}_{h}\|_{h}\,\|{\bf v}\|_{2}. Using (7) we get the result.

5 Convergence of the scheme

We first recall the stability result that has been proven in zimm2 . We deduce from it an estimate on the Fourier transform of the computed velocity (lemma 1). Using a result on space ${\bf P}_{0}$, we infer from it the convergence of the scheme (theorem 5.2). One shows that if the data ${\bf u}_{0}$ et ${\bf f}$ fulfill a compatibility condition heywood , there exists a solution $(\overline{\bf u},\overline{p})$ to equations (1)–(2) such that $\overline{\bf u}\in{\cal{C}}(0,T;{\bf H}^{2})$ , $\nabla\overline{p}\in{\cal{C}}(0,T;{\bf L}^{2})$. We assume from now on that there exists $C>0$ such that

[TABLE]

Let us recall the following result zimm2 .

Theorem 5.1

We assume that the initial values of the scheme fulfill (HI). There exists a constant $C>0$ such that for all $m\in\{2,\dots,N\}$

[TABLE]

and

[TABLE]

From now on we set ${\tilde{\mathbf{u}}}^{1}_{h}={\bf u}^{1}_{h}$ for the sake of conveniance. One deduces from hypothesis (HI) zimm1 that $|p^{1}_{h}|\leq C$ and $\|{\tilde{\mathbf{u}}}^{1}_{h}\|_{h}\leq C.$ Now, let $\varepsilon=\max(h,k)$. We study the behaviour of the scheme as $\varepsilon\to 0$. We define the applications ${\bf u}_{\varepsilon}:\mathbb{R}\to{\bf P}_{0}$ , ${\tilde{\mathbf{u}}}_{\varepsilon}:\mathbb{R}\to{\bf P}_{0}$ , ${\tilde{\mathbf{u}}}^{c}_{\varepsilon}:\mathbb{R}\to{\bf P}_{0}$, $p_{\varepsilon}:\mathbb{R}\to P^{nc}_{1}$ and ${\bf f}_{\varepsilon}:\mathbb{R}\to{\bf P}_{0}$ as follows. For all $n\in\{0,\dots,N-1\}$ and all $t\in[t_{n},t_{n+1}]$ we set

[TABLE]

and for all $t\not\in[0,T]$ we set ${\bf u}_{\varepsilon}(t)={\tilde{\mathbf{u}}}_{\varepsilon}(t)={\tilde{\mathbf{u}}}^{c}_{\varepsilon}(t)={\bf f}_{\varepsilon}(t)=\mathbf{0}$, $p_{\varepsilon}(t)=0$. We recall that the Fourier transform $\widehat{\bf v}$ of a function ${\bf v}\in{\bf L}^{1}(\mathbb{R})$ is defined by

[TABLE]

We have the following result.

Lemma 1

Let $0<\gamma<\frac{1}{4}$. There exists $C>0$ such that for all $\varepsilon>0$

[TABLE]

Proof. Let $\chi_{I}$ be the characteristic function of an interval $I\subset\mathbb{R}$. We define the application ${\bf g}_{\varepsilon}:\mathbb{R}\to{\bf P}_{0}$ as follows. For all $t\not\in[t_{1},T]$ we set ${\bf g}_{\varepsilon}(t)=\mathbf{0}$. For all $t\in[t_{1},T]$, ${\bf g}_{\varepsilon}(t)\in{\bf P}_{0}$ is the solution of

[TABLE]

with ${\bf P}_{\varepsilon}=-\nabla_{h}p_{\varepsilon}-2\,\frac{{\tilde{\mathbf{u}}}_{\varepsilon}(t-k)-{\bf u}_{\varepsilon}}{k}+\frac{{\tilde{\mathbf{u}}}_{\varepsilon}(t-2\,k)-{\bf u}_{\varepsilon}(t-k)}{2\,k}\,\chi_{[t_{2},T]}$. We have omitted most of the time dependancies for the sake of concision. Let us estimate ${\bf g}_{\varepsilon}$. We have

[TABLE]

According to proposition 4 we have

[TABLE]

Using the Cauchy-Schwarz inequality and (6) we have

[TABLE]

According to proposition 2 and theorem 5.1

[TABLE]

Using (18) we have

[TABLE]

Using proposition 3 and the Cauchy-Schwarz inequality we get

[TABLE]

Using proposition 5 we have

[TABLE]

Let us plug these estimates into (20). By simplifying by $\|{\bf g}_{\varepsilon}\|_{h}$ and integrating from $t=t_{1}$ to $T$ we get

[TABLE]

According to the Cauchy-Schwarz inequality and theorem 5.1

[TABLE]

Thanks to the stability of $\Pi_{{\bf P}_{0}}$ for the ${\bf L}^{2}$ norm we have

[TABLE]

And thanks to the Cauchy-Schwarz inequality and theorem 5.1

[TABLE]

Thus, since ${\bf g}_{\varepsilon}(t)=0$ for $t\in[0,t_{1}]$, we get $\int^{T}_{0}\|{\bf g}_{\varepsilon}(t)\|_{h}=\int^{T}_{t_{1}}\|{\bf g}_{\varepsilon}(t)\|_{h}\,dt\leq C$. Using definition (19) we obtain finally

[TABLE]

With this estimate we can now prove the result. Since the function ${\tilde{\mathbf{u}}}^{c}_{\varepsilon}$ is piecewise ${\cal{C}}^{1}$ on $\mathbb{R}$, and discontinous for $t=0$ and $t=T$, equation (• ‣ 3) reads

[TABLE]

where $\delta_{0}$, $\delta_{t_{1}}$, $\delta_{T}$ and $\delta_{T+k}$ are Dirac distributions located respectively in [math], $t_{1}$, $T$ and $T+k$. Let $\tau\in\mathbb{R}$. Applying the Fourier transform we get

[TABLE]

with

[TABLE]

Taking the scalar product with $i\,\widehat{\tilde{\mathbf{u}}}_{\varepsilon}(\tau)$ we get

[TABLE]

Let us bound the right-hand side. According to proposition 4 and (21)

[TABLE]

On the other hand, using theorem 5.1, one checks that $\alpha$ ​​ is bounded. Thus, according to the Cauchy-Schwarz inequality and (6)

[TABLE]

Hence we have

[TABLE]

If $\tau\neq 0$, multiplying this estimate by $|\tau|^{2\,\gamma-1}$, we get $|\tau|^{2\,\gamma}\,|\widehat{\tilde{\mathbf{u}}}_{\varepsilon}(\tau)|^{2}\leq C\,|\tau|^{2\,\gamma-1}\,\|\widehat{\tilde{\mathbf{u}}}_{\varepsilon}(\tau)\|_{h}$. Using the Young inequality and integrating over $\{\tau\in\mathbb{R}\,;|\tau|>1\}$ we obtain

[TABLE]

For $|\tau|\leq 1$ we have $|\tau|^{2\,\gamma}\,|\widehat{\tilde{\mathbf{u}}}_{\varepsilon}(\tau)|^{2}\leq|\widehat{\tilde{\mathbf{u}}}_{\varepsilon}(\tau)|^{2}\leq C\,\|\widehat{\tilde{\mathbf{u}}}_{\varepsilon}(\tau)\|^{2}_{h}$ thanks to (6). Thus

[TABLE]

Since $4\,\gamma-2<-1$ we have $\int_{|\tau|>1}|\tau|^{4\,\gamma-2}\,d\tau\leq C$. On the other hand, thanks to the Parseval theorem and thorem 5.1

[TABLE]

Hence the result.

We introduce the following spaces

[TABLE]

We also set

[TABLE]

for all ${\bf u}=(u_{1},u_{2})\in{\bf H}^{1}$, ${\bf v}=(v_{1},v_{2})\in{\bf H}^{1}$, ${\bf w}=(w_{1},w_{2})\in{\bf H}^{1}$. We have the following result.

Theorem 5.2

We assume that the initial values of the scheme fulfill hypothesis (HI). We also assume that the space step $h$ and the time step $k$ are such that $h\leq C\,k^{\alpha}$ with $\alpha>1$. Then we have ${\bf u}_{\varepsilon}\to{\bf u}$ in $L^{2}(0,T;{\bf L}^{2})$ with

[TABLE]

We also have ${\bf u}(0)={\bf u}_{0}$ and for all $\psi\in{\cal{C}}^{\infty}_{0}([0,T])$

[TABLE]

Proof. In what follows, sub-sequences of a sequence $({\bf v}_{\varepsilon})_{\varepsilon>0}$ will still be noted $({\bf v}_{\varepsilon})_{\varepsilon>0}$ for the sake of convenience. All the limits are for $\varepsilon\to 0$. According to theorem 5.1 and hypothesis (HI) we have

[TABLE]

We also deduce from (6), hypothesis (HI) and theorem 5.1

[TABLE]

A simple computation shows that there exists $C>0$ such that

[TABLE]

Thus the sequences $({\bf u}_{\varepsilon})_{\varepsilon>0}$, $({\tilde{\mathbf{u}}}_{\varepsilon})_{\varepsilon>0}$ and $({\tilde{\mathbf{u}}}^{c}_{\varepsilon})_{\varepsilon>0}$ are bounded in $L^{2}(0,T;{\bf L}^{2})$. Therefore there exists ${\bf u}\in L^{2}(0,T;{\bf L}^{2})$, ${\tilde{\mathbf{u}}}\in L^{2}(0,T;{\bf L}^{2})$ and ${\tilde{\mathbf{u}}}^{c}\in L^{2}(0,T;{\bf L}^{2})$ such that, up to a sub-sequence, we have

[TABLE]

We claim that the limits ${\bf u}$, ${\tilde{\mathbf{u}}}$, ${\tilde{\mathbf{u}}}^{c}$ are the same. Indeed, let us consider ${\bf u}_{\varepsilon}-{\tilde{\mathbf{u}}}_{\varepsilon}$. Since ${\bf u}^{n}_{h}-{\tilde{\mathbf{u}}}^{n+1}_{h}=({\bf u}^{n}_{h}-{\bf u}^{n+1}_{h})+({\bf u}^{n+1}_{h}-{\tilde{\mathbf{u}}}^{n+1}_{h})$ we have

[TABLE]

According to theorem 5.1 we have $k\sum_{n=0}^{N-1}|{\bf u}^{n}_{h}-{\bf u}^{n+1}_{h}|^{2}\leq C\sum_{n=0}^{N-1}k^{3}\leq C\,k^{2}$. Thanks to (18) we also have

[TABLE]

Thus $\|{\bf u}_{\varepsilon}-{\tilde{\mathbf{u}}}_{\varepsilon}\|_{L^{2}(0,T;{\bf L}^{2})}\to 0$ and ${\bf u}={\tilde{\mathbf{u}}}$. One checks in a simililar way that ${\tilde{\mathbf{u}}}={\tilde{\mathbf{u}}}^{c}$. Now, using the Fourier transform, we prove the strong convergence of the sequence $({\bf u}_{\varepsilon})_{\varepsilon>0}$ in $L^{2}(0,T;{\bf L}^{2})$. We set ${\bf v}_{\varepsilon}={\bf u}_{\varepsilon}-{\bf u}$. Let $M>0$. We use the splitting

[TABLE]

Let us estimate $J^{M}_{\varepsilon}$. Since $|\widehat{{\bf v}}_{\varepsilon}(\tau)|^{2}\leq 2\,|\widehat{{\bf u}}_{\varepsilon}(\tau)|^{2}+2\,|\widehat{{\bf u}}(\tau)|^{2}$ we have

[TABLE]

According to lemma 1 we have

[TABLE]

Thus

[TABLE]

Therefore, for all $\varepsilon>0$, we have $J^{M}_{\varepsilon}\to 0$ when $M\to\infty$. We now consider $I^{M}_{\varepsilon}$. Let $\tau\in\mathbb{R}$. Since ${\bf u}_{\varepsilon}\rightharpoonup{\bf u}$ in $L^{2}(0,T;{\bf L}^{2})$, we deduce from definition (19)

[TABLE]

For all $t\in\mathbb{R}$ we have ${\tilde{\mathbf{u}}}_{\varepsilon}(t)\in{\bf P}_{0}$. From definition (19) we infer that $\widehat{{\tilde{\mathbf{u}}}}_{\varepsilon}(\tau)\in{\bf P}_{0}$. Now, prolonging $\widehat{{\tilde{\mathbf{u}}}}_{\varepsilon}(\tau)$ by [math] outside $\Omega$, we deduce from lemma 4 in eymgal that there exists a constant $C>0$ such that

[TABLE]

Using definition (19), the Cauchy-Schwarz inequality and theorem 5.1, we have

[TABLE]

Thus, using the compactness criterium given by theorem 1 in eymgal , we get $\widehat{{\tilde{\mathbf{u}}}}_{\varepsilon}(\tau)\to\widehat{{\bf u}}(\tau)$ in ${\bf L}^{2}$. Thus $\widehat{\widetilde{{\bf v}}}_{\varepsilon}(\tau)=\widehat{{\tilde{\mathbf{u}}}}_{\varepsilon}(\tau)-\widehat{{\bf u}}(\tau)\to 0$ in ${\bf L}^{2}$. Therefore for all $M>0$ we have $I^{M}_{\varepsilon}\to 0$. Using the Parseval inequality, and gathering the limits for $I^{M}_{\varepsilon}$ and $J^{M}_{\varepsilon}$, we get

[TABLE]

We have proven that ${\bf u}_{\varepsilon}\to{\bf u}$ in $L^{2}(0,T;{\bf L}^{2})$.

We now check the properties of ${\bf u}$. First, proceeding as in eymgal , one checks easily that ${\bf u}\in L^{2}(0,T;{\bf H}^{1}_{0})$. Now let $q\in L^{2}(0,T;{\cal{C}}^{\infty}_{0})$. According to (12) we have $\nabla_{h}(\Pi_{P^{nc}_{1}}q)\to\nabla q$ in $L^{2}(0,T;{\bf L}^{2})$. Since ${\bf u}_{\varepsilon}\to{\bf u}$ in $L^{2}(0,T;{\bf L}^{2})$ we get

[TABLE]

On the other hand, according to propositions 1 and 3, we have for all $\varepsilon>0$

[TABLE]

Thus we have $\int^{T}_{0}q\,\hbox{div}\,{\bf u}\,dt=0$ for all $q\in L^{2}(0,T;{\cal{C}}^{\infty}_{0})$. Since the space ${\cal{C}}^{\infty}_{0}$ is dense in $L^{2}$, we get $\hbox{div}\,{\bf u}=0$. Hence ${\bf u}\in L^{2}(0,T;\mathbf{V})$. Let us now check the regularity of $\displaystyle\frac{d{\bf u}}{dt}$. Using hypothesis (HI), (6) and theorem 5.1, we have

[TABLE]

Thus the sequence $\displaystyle\left(\frac{d{\tilde{\mathbf{u}}}^{c}_{\varepsilon}}{dt}\right)_{\varepsilon>0}$ is bounded in $L^{2}(0,T;{\bf L}^{2})$. Since ${\bf u}_{\varepsilon}\to{\bf u}$ in $L^{2}(0,T;{\bf L}^{2})$ with ${\bf u}\in L^{2}(0,T;{\bf H}^{1})$, proceeding as in, we get

[TABLE]

and ${\bf u}\in{\mathcal{C}}(0,T;{\bf H})$.

Let us now prove that ${\bf u}$ satisfies (23). For the sake of simplicity, we omit to note some time dependencies. According to (• ‣ 3) we have for all $t\in[t_{1},T]$

[TABLE]

Let ${\bf v}\in\mathbf{V}\cap({\cal{C}}^{\infty}_{0})^{2}$ and $\psi\in{\cal{C}}^{\infty}([0,T])$ with $\psi(T)=0$. We set ${\bf v}_{h}=\widetilde{\Pi}_{{\bf P}_{0}}{\bf v}$. Multiplying the former equation by $\psi\,{\bf v}_{h}$ and integrating over $[t_{1},T]$ we get

[TABLE]

with $\chi=-\chi_{[t_{3},T-2\,k]}+\frac{1}{3}\,\chi_{[t_{2},t_{3}]}-\frac{4}{3}\,\chi_{[t_{1},t_{2}]}-\frac{7}{3}\,\chi_{[T-k,T]}-\frac{2}{3}\,\chi_{[T-2\,k,T-k]}$. We now check the limits of the terms in this equation. First, according to (9), we have ${\bf v}_{h}\to{\bf v}$ in ${\bf L}^{2}$. We will use this limit in the computations below without mentioning it. Since $\psi(T)=0$ we obtain by integrating by parts

[TABLE]

and

[TABLE]

According to hypothesis (HI) we have ${\tilde{\mathbf{u}}}^{0}_{h}={\bf u}^{0}_{h}\to{\bf u}_{0}$ in ${\bf L}^{2}$ and ${\tilde{\mathbf{u}}}^{1}_{h}={\bf u}^{1}_{h}\to{\bf u}_{0}$ in ${\bf L}^{2}$. It implies that $({\bf u}^{0}_{h},{\bf v}_{h})\to({\bf u}_{0},{\bf v})$ and $\psi(t_{1})\,({\tilde{\mathbf{u}}}^{1}_{h},{\bf v}_{h})=\psi(k)\,({\tilde{\mathbf{u}}}^{1}_{h},{\bf v}_{h})\to\psi(0)\,({\bf u}_{0},{\bf v})$. On the other hand

[TABLE]

and since $\chi_{[0,T-k]}\,\psi^{\prime}(\cdot+k)\to\psi^{\prime}$ in $L^{\infty}(0,T)$

[TABLE]

Thus we have

[TABLE]

Let us now consider the discrete laplacian. Using proposition 6 and the splitting {\mathbf{\widetilde{\Delta}}}_{h}{\bf v}_{h}=\big{(}{\mathbf{\widetilde{\Delta}}}_{h}{\bf v}_{h}-\Pi_{{\bf P}_{0}}({\mathbf{\widetilde{\Delta}}}{\bf v})\big{)}+\Pi_{{\bf P}_{0}}({\mathbf{\widetilde{\Delta}}}{\bf v}) we have

[TABLE]

with A_{\varepsilon}=\int^{T}_{0}\psi\,\big{(}{\tilde{\mathbf{u}}}_{\varepsilon},{\mathbf{\widetilde{\Delta}}}_{h}(\widetilde{\Pi}_{{\bf P}_{0}}{\bf v})-\Pi_{{\bf P}_{0}}({\mathbf{\widetilde{\Delta}}}{\bf v})\big{)}\,dt, B_{\varepsilon}=\int^{T}_{0}\psi\,\big{(}{\tilde{\mathbf{u}}}_{\varepsilon},\Pi_{{\bf P}_{0}}(\Delta{\bf v})\big{)}\,dt. Since

[TABLE]

using proposition 7 and the Cauchy-Schwarz inequality, we get

[TABLE]

Therefore, using theorem 5.1: $|A_{\varepsilon}|\leq C\,h\left(k\sum_{n=1}^{N}\|{\tilde{\mathbf{u}}}^{n}_{h}\|^{2}_{h}\right)^{1/2}\leq C\,h$. Hence $A_{\varepsilon}\to 0$. On the other hand, using an integration by parts, we have

[TABLE]

By gathering the limits for $A_{\varepsilon}$ and $B_{\varepsilon}$ we get

[TABLE]

Let us now consider the pressure. We use the splitting

[TABLE]

First, integrating by parts, we have

[TABLE]

Since $\hbox{div}\,{\bf v}=0$, using the divergence formula and definition (13), one checks that $\hbox{div}\,(\Pi_{\mathbf{RT_{0}}}{\bf v})=0$. Thus -\big{(}p_{\varepsilon},\hbox{div}\,(\Pi_{\mathbf{RT_{0}}}{\bf v})\big{)}=0. On the other hand

[TABLE]

and since $p_{\varepsilon}\in P^{nc}_{1}$ we get $\sum_{K\in{\cal{T}}_{h}}\int_{\partial K}p_{\varepsilon}\,(\Pi_{\mathbf{RT_{0}}}{\bf v}\cdot{\mathbf{n}}_{K,\sigma})=0$. Thus the last term in (26) vanishes. To bound the other terms, we use the Cauchy-Schwartz inequality together with estimates (9), (14) and theorem 5.1. We get

[TABLE]

Plugging these estimates into (26) we get $\int^{T}_{t_{1}}\left|\chi\,\psi\,(\nabla_{h}p_{\varepsilon},{\bf v}_{h})\right|\,dt\leq C\,\frac{h}{k}$. By hypothesis we have $\frac{h}{k}\leq k^{\alpha-1}$ with $\alpha-1>0$. Thus for $\varepsilon=\max(h,k)\to 0$

[TABLE]

Let us now consider the convection term. We set $\overline{\bf u}_{\varepsilon}=2\,{\bf u}_{\varepsilon}-{\bf u}_{\varepsilon}(t-k)$ and want to find the limit of $\int^{T}_{t_{1}}\psi\,{\hbox{\rm b}}_{h}(\overline{\bf u}_{\varepsilon},{\tilde{\mathbf{u}}}_{\varepsilon},{\bf v}_{h})\,dt$. We use the splitting $-{\hbox{\rm b}}_{h}(\overline{\bf u}_{\varepsilon},{\tilde{\mathbf{u}}}_{\varepsilon},{\bf v}_{h})+{\hbox{\rm b}}({\bf u},{\bf u},{\bf v})=A^{\varepsilon}_{1}+A^{\varepsilon}_{2}+A^{\varepsilon}_{3}$ with

[TABLE]

[TABLE]

By definition A^{\varepsilon}_{1}=-\sum_{i=1}^{2}\big{(}u_{i},({\bf u}-\overline{\bf u}_{\varepsilon})\cdot\nabla v_{i}\big{)}. Using the Cauchy-Schwarz inequality we get

[TABLE]

Since ${\bf u}_{\varepsilon}\to{\bf u}$ in $L^{2}(0,T;{\bf L}^{2})$ we also have $\|{\bf u}-\overline{\bf u}_{\varepsilon}\|_{L^{2}(0,T;{\bf L}^{2})}\to 0$. Thus $\int^{T}_{t_{1}}\psi\,A^{\varepsilon}_{1}\,dt\to 0$. Let us now consider $A^{\varepsilon}_{2}$. Since $\overline{\bf u}_{\varepsilon}\cdot{\mathbf{n}}|_{\partial\Omega}=0$ we obtain by integrating by parts {\hbox{\rm b}}(\overline{\bf u}_{\varepsilon},{\bf u},{\bf v})=\sum_{i=1}^{2}\big{(}v_{i},\hbox{div}(u_{i}\,\overline{\bf u}_{\varepsilon})\big{)}. Thus

[TABLE]

Using the Cauchy-Schwarz inequality we get

[TABLE]

Using a Taylor expansion one checks that $\|{\bf v}-{\bf v}_{h}\|_{{\bf L}^{\infty}}\leq\|{\bf v}\|_{\mathbf{W}^{1,\infty}}\,h$. We recall also that $\|{\bf u}_{\varepsilon}\|_{L^{2}(0,T;{\bf L}^{2})}\leq C$. Therefore $\int^{T}_{t_{1}}\psi\,A^{\varepsilon}_{2}\,dt\to 0$. Let us now bound $A^{\varepsilon}_{3}$. For all triangle $K\in{\cal{T}}_{h}$ and all edge $\sigma\in{\cal{E}}_{K}\cap{\cal{E}}^{int}_{h}$, we set

[TABLE]

Using the divergence formula one checks that

[TABLE]

By writing this sum as a sum on the edges we get

[TABLE]

Thus, using definition (11) and a quadrature formula

[TABLE]

We have $|\sigma|\leq h$ and, using a Taylor expansion, one checks that $|{\bf v}_{K_{\sigma}}-{\bf v}_{L_{\sigma}}|\leq h\,\|{\bf v}\|_{\mathbf{W}^{1,\infty}}$. Thus, thanks to the Cauchy-Schwarz inequality, we get

[TABLE]

Using (4) we get

[TABLE]

Therefore, using a quadrature formula

[TABLE]

By writing $\Pi_{{\bf P}^{nc}_{1}}{\bf u}-{\bf u}_{\varepsilon}=(\Pi_{{\bf P}^{nc}_{1}}{\bf u}-{\bf u})+({\bf u}-{\bf u}_{\varepsilon})$ and using (12), we get $|A^{\varepsilon}_{3}|\leq C\,|{\bf u}_{\varepsilon}|\,(h\,\|{\bf u}\|_{1}+|{\bf u}-{\bf u}_{\varepsilon}|)$. Thus, using the Cauchy-Schwarz inequality, we have

[TABLE]

Thus $\int^{T}_{t_{1}}\psi\,A^{\varepsilon}_{3}\,dt\to 0$. By gathering the limits for $A^{\varepsilon}_{1}$, $A^{\varepsilon}_{2}$, $A^{\varepsilon}_{3}$ we obtain \int^{T}_{t_{1}}\psi\,\big{(}{\hbox{\rm b}}_{h}(\overline{\bf u}_{\varepsilon},{\bf u}_{\varepsilon},{\bf v}_{h})-{\hbox{\rm b}}({\bf u},{\bf u},{\bf v})\big{)}\,dt\to 0. Since $\int^{t_{1}}_{0}{\hbox{\rm b}}({\bf u},{\bf u},{\bf v})\,dt\to 0$, we get

[TABLE]

Finally, since ${\bf v}_{h}\in{\bf P}_{0}$, we have: $({\bf f}_{\varepsilon},{\bf v}_{h})=(\Pi_{{\bf P}_{0}}{\bf f},{\bf v}_{h})=({\bf f},{\bf v}_{h})$. Therefore

[TABLE]

We now gather the limits we have obtained into (5). The space $\mathbf{V}\cap({\cal{C}}^{\infty}_{0})^{2}$ is dense in $\mathbf{V}$. Hence we obtain for all ${\bf v}\in\mathbf{V}$ and $\psi\in{\cal{C}}^{\infty}([0,T])$ with $\psi(T)=0$

[TABLE]

Taking $\psi=\phi\in{\cal{C}}^{\infty}_{0}([0,T])$, we have $\phi(0)=0$ and from the definition of the derivative in the distributional sense

[TABLE]

Thus we have proven (23). At last, let us show that the initial condition holds. We have proven before that

[TABLE]

Let ${\bf v}\in\mathbf{V}\cap({\cal{C}}^{\infty}_{0})^{2}$ and $\psi\in{\cal{C}}^{\infty}([0,T])$ such that $\psi(T)=0$. We have

[TABLE]

Integrating by parts the limit we get

[TABLE]

By comparing this limit with (25), we get $\psi(0)\,({\bf u}(0)-{\bf u}_{0},{\bf v})=0$ for all $\psi\in{\cal{C}}^{\infty}([0,T])$ with $\psi(T)=0$. Therefore ${\bf u}(0)={\bf u}_{0}$. At last, note that we have proven so far the convergence of a sub-sequence of $({\bf u}_{\varepsilon})_{\varepsilon>0}$ towards ${\bf u}$. But the application ${\bf u}$ such that (22), (23) and ${\bf u}(0)={\bf u}_{0}$ hold is unique (temam , p. 254). Thus the whole sequence $({\bf u}_{\varepsilon})_{\varepsilon>0}$ converges towards ${\bf u}$.