A Weak Galerkin Mixed Finite Element Method for second order elliptic equations on 2D Curved Domains

TL;DR

This paper analyzes the weak Galerkin mixed finite element method for second order elliptic equations on curved 2D domains, identifies accuracy loss due to domain approximation, and proposes techniques to restore optimal convergence.

Contribution

It provides a detailed error analysis of WG-MFEM on curved domains and introduces boundary correction and mesh refinement techniques to achieve optimal accuracy.

Findings

Original WG-MFEM exhibits $O(h^{1/2})$ convergence on curved domains.

Boundary correction and mesh techniques restore optimal convergence rates.

Numerical results confirm theoretical error estimates.

Abstract

This article concerns the weak Galerkin mixed finite element method (WG-MFEM) for second order elliptic equations on 2D domains with curved boundary. The Neumann boundary condition is considered since it becomes the essential boundary condition in this case. It is well-known that the discrepancy between the curved physical domain and the polygonal approximation domain leads to a loss of accuracy for discretization with polynomial order $\alpha>1$. The purpose of this paper is two-fold. First, we present a detailed error analysis of the original WG-MFEM for solving problems on curved domains, which exhibits an $O(h^{1/2})$ convergence for all $\alpha\ge 1$. It is a little surprising to see that even the lowest-order WG-MFEM ($\alpha=1$) experiences a loss of accuracy. This is different from known results for the finite element method (FEM) or the mixed FEM, and appears to be a combined effect of the WG-MFEM design and the fact that the outward normal vector on the polygonal approximation domain is different from the one on the curved domain. Second, we propose a remedy to bring the approximation rate back to optimal by employing two techniques. One is a specially designed boundary correction technique. The other is to take full advantage of the nice feature that weak Galerkin discretization can be defined on polygonal meshes, which allows the curved boundary to be better approximated by multiple short edges without increasing the total number of mesh elements. Rigorous analysis shows that a combination of the above two techniques renders optimal convergence for all $\alpha$. Numerical results further confirm this conclusion.