Convergence analysis of a family of ELLAM schemes for a fully coupled model of miscible displacement in porous media

Abstract

We analyse the convergence of numerical schemes in the GDM–ELLAM (Gradient Discretisation Method–Eulerian Lagrangian Localised Adjoint Method) framework for a strongly coupled elliptic-parabolic PDE which models miscible displacement in porous media. These schemes include, but are not limited to, Mixed Finite Element–ELLAM and Hybrid Mimetic Mixed–ELLAM schemes. A complete convergence analysis is presented on the coupled model, using only weak regularity assumptions on the solution (which are satisfied in practical applications), and not relying on $L^\infty$ bounds (which are impossible to ensure at the discrete level given the anisotropic diffusion tensors and the general grids used in applications).

Appendix: generic compactness results

The following results are particular cases of more general theorems on GDM that can be found in [16].

Lemma 9.1

(Regularity of the limit, space–time problems [16, Lemma 4.8]) Let $p\in (1,\infty )$, and $(({\mathcal D}^T)_m)_{m\in \mathbb {N}}$ be a coercive and limit-conforming sequence of space–time GDs. For each $m\in \mathbb {N}$, take $u_m\in X_{{\mathcal D}_m}^{N_m+1}$ (identified with a piecewise-constant function $[0,T]\rightarrow X_{{\mathcal D}_m}$) and assume that $(\Vert u_m \Vert _{L^p(0,T;X_{{\mathcal D}_m})})_{m\in \mathbb {N}}$ is bounded. Then there exists $u\in L^p(0,T;H^1(\Omega ))$ such that, up to a subsequence as $m\rightarrow \infty$, $\Pi _{{\mathcal D}_m}u_m \rightarrow u$ and $\nabla _{{\mathcal D}_m}u_m \rightarrow \nabla u$ weakly in $L^p(0,T;L^2(\Omega ))$. The same property holds with $p=+\infty$, provided that the weak convergences are replaced by weak-$*$ convergences.

Definition 9.2

(Compactly–continuously embedded sequence) Let $(X_m,\Vert \cdot \Vert _{X_m})_{m\in \mathbb {N}}$ be a sequence of Banach spaces included in $L^2(\Omega )$, and $(Y_m,\Vert \cdot \Vert _{Y_m})_{m\in \mathbb {N}}$ be a sequence of Banach spaces. The sequence $(X_m,Y_m)_{m\in \mathbb {N}}$ is compactly–continuously embedded in $L^2(\Omega )$ if:

• (1) If $u_m\in X_m$ for all $m\in \mathbb {N}$ and $(\Vert u_m \Vert _{X_m})_{m\in \mathbb {N}}$ is bounded, then $(u_m)_{m\in \mathbb {N}}$ is relatively compact in $L^2(\Omega )$.

• (2) $X_m\subset Y_m$ for all $m\in \mathbb {N}$ and for any sequence $(u_m)_{m\in \mathbb {N}}$ such that

  • (a) $u_m\in X_m$ for all $m\in \mathbb {N}$ and $(\Vert u_m \Vert _{X_m})_{m\in \mathbb {N}}$ is bounded,

  • (b) $\Vert u_m \Vert _{Y_m}\rightarrow 0$ as $m \rightarrow \infty$,

  • (c) $(u_m)_{m\in \mathbb {N}}$ converges in $L^2(\Omega )$,

  it holds that $u_m\rightarrow 0$ in $L^2(\Omega )$.

Theorem 9.3

(Discrete Aubin–Simon compactness [16, Theorem C.8]) Let $(X_m, Y_m)_{m\in \mathbb {N}}$ be compactly–continuously embedded in $L^2(\Omega )$, $T > 0$ and $(f_m)_{m\in \mathbb {N}}$ be a sequence in $L^2(0,T;L^2(\Omega ))$ such that

• For all $m \in N$, there exists $N\in \mathbb {N}^*$, $0=t^{(0)}<\dots <t^{(N)}=T$ and $(v^{(n)})_{n=0,\dots ,N} \in X_{m}^{N+1}$ such that $f_m(t)=v^{(n+1)}$   for all $n=0,\dots ,N-1$ and a.e. $t\in (t^{(n)},t^{(n+1)}), f_m(t)=v^{(n+1)}$. We then set

  $$\begin{aligned} \delta _m f_m(t)= \dfrac{v^{(n+1)}-v^{(n)}}{t^{(n+1)}-t^{(n)}} \hbox { for }n=0,\dots ,N-1\hbox { and }t\in (t^{(n)},t^{(n+1)}). \end{aligned}$$

• The sequence $(f_m)_{m\in \mathbb {N}}$ is bounded in $L^2(0, T;L^2(\Omega ))$.

• The sequence $(\Vert f_m \Vert _{L^2(0,T;X_m)})_{m\in \mathbb {N}}$ is bounded.

• The sequence $(\Vert \delta _mf_m \Vert _{L^2(0,T;Y_m)})_{m\in \mathbb {N}}$ is bounded.

Then $(f_m)_{m\in \mathbb {N}}$ is relatively compact in $L^2(0,T;L^2(\Omega ))$.