Superconvergence of MAC Scheme for a Coupled Free Flow-Porous Media System with Heat Transport on Non-uniform Grids

Abstract

In this paper, a marker and cell (MAC) scheme is constructed for the free flow-porous media system with heat transport on non-uniform grids, where the Stokes-Darcy equation is employed to describe the free-flow and the porous regions. Error estimates for the velocity, pressure and temperature in different discrete norms are established rigorously and carefully by constructing discrete auxiliary functions. We obtain the second order superconvergence in the discrete $L^2$ norm for velocity, pressure and temperature and the second order superconvergence for some terms of the $H^1$ norm for the velocity on non-uniform grids. Several numerical examples verifying the theoretical results and illustrating the capabilities of the method are demonstrated.

Appendix A Preliminaries for the Discrete LBB Condition

In this appendix, we give some details of the discrete LBB condition. Here we use notation and results similar to [26, 28]. Let

$$\begin{aligned} b(\mathbf{v} ,q)=-\int _\Omega ~qdiv \mathbf{v} dx, ~\mathbf{v} \in \mathbf{V} ,~q\in W, \end{aligned}$$

where $\mathbf{V} = \mathbf{V} _f \cup \mathbf{V} _m$ with norm for the velocity

$$\begin{aligned} \Vert \mathbf{v} \Vert _{\mathbf{V} } = ( \Vert \nabla \mathbf{v} \Vert _{\Omega _f}^2 +\Vert \mathbf{v} \Vert _{\Omega _m}^2 + \Vert \nabla \cdot \mathbf{v} \Vert _{\Omega _m}^2 )^{1/2}. \end{aligned}$$

Here

$$\begin{aligned}&\mathbf{V} _f := \{\mathbf{v } \in ( H^1(\Omega _f) )^2 , \mathbf{v} |_{\Gamma _f}=0 \},\ \mathbf{V} _m := \{\mathbf{v } \in H(div;\Omega _m), \ \mathbf{v} \cdot \mathbf{n} |_{\Gamma _m} =0 \},\\&W= \left\{ q \in W_f \cup W_m, \int _{\Omega } q dx=0 \right\} , \ W_f := L^2(\Omega _f), \ W_m := L^2(\Omega _m). \end{aligned}$$

with $H(div;\Omega _m)$ as a Hilbert space defined by

$$\begin{aligned} H(div;\Omega _m) = \{ \mathbf{v} \in ( L^2(\Omega _m) )^2; \nabla \cdot \mathbf{v} \in L^2( \Omega _m ) \}, \end{aligned}$$

Fig. 3

Partitions: a ${\mathcal {T}}_h$, b ${\mathcal {T}}_h^1$, c ${\mathcal {T}}_h^2$

Then we construct the finite-dimensional subspaces of W and $\mathbf{V}$ by introducing three different partitions ${\mathcal {T}}_h,{\mathcal {T}}_h^1,{\mathcal {T}}_h^2$ of $\Omega$. The original partition $\delta _x\times \delta _y$ is denoted by ${\mathcal {T}}_h$ (see Fig. 3). The partition ${\mathcal {T}}_h^1$ is generated by connecting all the midpoints of the vertical sides of $\Omega _{i+1/2,j+1/2}$ and extending the resulting mesh to the boundary $\Gamma$. Similarly, for all $\Omega _{i+1/2,j+1/2}\in {\mathcal {T}}_h$ we connect all the midpoints of the horizontal sides of $\Omega _{i+1/2,j+1/2}$ and extend the resulting mesh to the boundary $\Gamma$, then the third partition is obtained which is denoted by ${\mathcal {T}}_h^2$.

Corresponding to the quadrangulation ${\mathcal {T}}_h$, define $W_h$, a subspace of W,

$$\begin{aligned} W_h=\left\{ q_h:~q_h|_T=constant,~\forall T\in {\mathcal {T}}_h ~and \int _\Omega qdx=0\right\} . \end{aligned}$$

Furthermore, let $\mathbf{V} _h$ be a subspace of $\mathbf{V}$ such that $\mathbf{V} _h= \mathbf{V} \cap ( \mathbf{V} _{f,h} \cup \mathbf{V} _{m,h})$ equipped with the discrete norm for velocity

$$\begin{aligned} \Vert \mathbf{v} \Vert _h= ( \Vert D\mathbf{v} \Vert _f^2 + \Vert \mathbf{v} \Vert _m^2 )^{1/2}, \end{aligned}$$

where

$$\begin{aligned} \mathbf{V} _{f,h}&:= \{ \mathbf{v} = (v^x,v^y) \in C^{(0)} (\overline{\Omega } _f )^2: v^x|_{T^1} \in Q_1(T^1), \ \forall T^1\in {\mathcal {T}}_h^1 \cap \Omega _f ; \\&\qquad v^y |_{T^2} \in Q_1(T^2), \ \forall T^2 \in {\mathcal {T}}_h^2 \cap \Omega _f \} \end{aligned}$$

and $Q_1$ denotes the space of all polynomials of degree $\le 1$ with respect to each of the two variables x and y. On the porous medium $\Omega _m$, define

$$\begin{aligned} \mathbf{V} _{m,h}&:= \{ \mathbf{v} = (v^x,v^y) \in H(div, \Omega _m ): v^x|_{T^1} \in Q_{1,0}(T^1), \ \forall T^1\in {\mathcal {T}}_h^1 \cap \Omega _m ; \\&\qquad v^y |_{T^2} \in Q_{0,1}(T^2), \ \forall T^2 \in {\mathcal {T}}_h^2 \cap \Omega _m \} \end{aligned}$$

and $Q_{m,s}$ denotes the space of all polynomials of degree $\le m$ with respect to x and degree $\le s$ with respect to y. It is clear that $\mathbf{V} _{m,h}$ is the lowest-order Raviart-Thomas-Nedelec (RTN) spaces on rectangles [1, 16].

Then we introduce the bilinear forms

$$\begin{aligned} b_h(\mathbf{v} _h,q_h)=-\sum _{\Omega _{i+1/2,j+1/2} \in {\mathcal {T}}_h}\int _{\Omega _{i+1/2,j+1/2}} q_h P_h(div\mathbf{v} _h)dx,~\mathbf{v} _h\in \mathbf{V} _h,~q_h\in W_h, \end{aligned}$$

where $P_h: \ C^{(0)}(\overline{\Omega }_{i+1/2,j+1/2}) \rightarrow Q_0(\Omega _{i+1/2,j+1/2})$ such that

$$\begin{aligned}&(P_h\varphi )_{i+1/2,j+1/2}=\varphi _{i+1/2,j+1/2}, ~~\forall ~\Omega _{i+1/2,j+1/2}\in {\mathcal {T}}_h. \end{aligned}$$