Convergence of the Marker-and-Cell Scheme for the Incompressible Navier–Stokes Equations on Non-uniform Grids

Abstract

We prove in this paper the convergence of the Marker-and-Cell scheme for the discretization of the steady-state and time-dependent incompressible Navier–Stokes equations in primitive variables, on non-uniform Cartesian grids, without any regularity assumption on the solution. A priori estimates on solutions to the scheme are proven; they yield the existence of discrete solutions and the compactness of sequences of solutions obtained with family of meshes the space step and, for the time-dependent case, the time step of which tend to zero. We then establish that the limit is a weak solution to the continuous problem.

Appendix: Discrete Functional Analysis

Definition 5.1

(Compactly embedded sequence of spaces) Let B be a Banach space; a sequence $(X_m)_{{m\in \mathbb {N}}}$ of Banach spaces included in B is compactly embedded in B if any sequence $(u_m)_{m\in \mathbb {N}}$ satisfying:

• $u_m\in X_m$ ($\forall m\in \mathbb {N}$),

• the sequence $(\Vert u_m\Vert _{X_m})_{{m\in \mathbb {N}}}$ is bounded,

is relatively compact in B.

Definition 5.2

(Compact-continuous sequence of spaces) Let B be a Banach space, and let $(X_m)_{{m\in \mathbb {N}}}$ and $(Y_m)_{{m\in \mathbb {N}}}$ be sequences of Banach spaces such that $X_m \subset B$ for $m\in \mathbb {N}$. The sequence $(X_m,Y_m)_{{m\in \mathbb {N}}}$ is compact-continuous in B if the following conditions are satified:

• The sequence $(X_m)_{{m\in \mathbb {N}}}$ is compactly embedded in B (see Definition 5.1),

• $X_m\subset Y_m$ (for all ${m\in \mathbb {N}}$),

• if the sequence $(u_m)_{{m\in \mathbb {N}}}$ is such that $u_m\in X_m$ (for all ${m\in \mathbb {N}}$), $(\Vert u_m\Vert _{X_m})_{{m\in \mathbb {N}}}$ is bounded and $\Vert u_m\Vert _{Y_m}\rightarrow 0$ as $m\rightarrow +\infty$, then any subsequence of $(u_m)_{{m\in \mathbb {N}}}$ converging in B converges to 0 (in B).

The following theorem is proved [4] and is a generalization of a previous work carried out in [16].

Theorem 5.3

(Aubin–Simon Theorem with a sequence of subspaces and a discrete derivative.) Let $1 \le p < \infty$, let B be a Banach space, and let $(X_m)_{{m\in \mathbb {N}}}$ and $(Y_m)_{{m\in \mathbb {N}}}$ be sequences of Banach spaces such that $X_m \subset B$ for $m\in \mathbb {N}$. We assume that the sequence $(X_m,Y_m)_{{m\in \mathbb {N}}}$ is compact-continuous in B. Let $T>0$ and $(u^{(m)})_{{m\in \mathbb {N}}}$ be a sequence of $L^{p}(0,T;B)$ satisfying the following conditions:

• (H1) the sequence $(u^{(m)})_{{m\in \mathbb {N}}}$ is bounded in $L^{p}(0,T;B)$.

• (H2) the sequence $(\Vert u^{(m)}\Vert _{L^{1}(0,T;X_m)})_{{m\in \mathbb {N}}}$ is bounded.

• (H3) the sequence $(\Vert \eth _t u^{(m)}\Vert _{L^{p}(0,T;Y_m)})_{{m\in \mathbb {N}}}$ is bounded.

Then there exists $u\in L^{p}(0,T;B)$ such that, up to a subsequence, $u^{(m)}\rightarrow u$ in $L^{p}(0,T;B)$.

Definition 5.4

(B -limit-included) Let B be a Banach space, $(X_m)_{m\in \mathbb {N}}$ be a sequence of Banach spaces included in B and X be a Banach space included in B. The sequence $(X_m)_{m\in \mathbb {N}}$ is B-limit-included in X if there exists $C\in \mathbb {R}$ such that if u is the limit in B of a subsequence of a sequence $(u_m)_{{m\in \mathbb {N}}}$ verifying $u_m\in X_m$ and $\Vert u_m\Vert _{X_m}\le 1$, then $u\in X$ and $\Vert u\Vert _{X}\le C$.

The regularity of a possible limit of approximate solutions may be proved thanks to the theorem which we recall below [17, Theorem B1].

Theorem 5.5

(Regularity of the limit) Let $1\le p< \infty$ and $T>0$. Let B be a Banach space, $(X_m)_{m\in \mathbb {N}}$ be a sequence of Banach spaces included in B and B-limit-included in X (where X is a Banach space included in B). Let $T>0$ and, for $m\in \mathbb {N}$, Let $u_m \in L^{p}(0,T;X_m)$. We assume that the sequence $(\Vert u_m\Vert _{L^{p}(0,T;X_m)})_{m\in \mathbb {N}}$ is bounded and that $u_m\rightarrow u$ a.e. as $m\rightarrow \infty$. Then, $u\in L^{p}(0,T;X)$.