A Weak Galerkin Mixed Finite Element Method for Second Order Elliptic Equations on 2D Curved Domains (Communications in Computational Physics, 2022)
This article concerns the weak Galerkin mixed finite element method (WG-MFEM) for second order elliptic equations on 2D domains with curved boundary. TheNeumann boundary condition is considered since it becomes the essential boundarycondition 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 fordiscretization with polynomial order $α>1.$ The purpose of this paper is two-fold. First,we present a detailed error analysis of the original WG-MFEM for solving problemson curved domains, which exhibits an $O(h^{1/2})$ convergence for all $α ≥ 1.$ It is a littlesurprising to see that even the lowest-order WG-MFEM $(α = 1)$ experiences a loss ofaccuracy. This is different from known results for the finite element method (FEM) orthe mixed FEM, and appears to be a combined effect of the WG-MFEM design andthe fact that the outward normal vector on the polygonal approximation domain isdifferent from the one on the curved domain. Second, we propose a remedy to bringthe approximation rate back to optimal by employing two techniques. One is a specially designed boundary correction technique. The other is to take full advantage ofthe nice feature that weak Galerkin discretization can be defined on polygonal meshes,which allows the curved boundary to be better approximated by multiple short edgeswithout increasing the total number of mesh elements. Rigorous analysis shows thata combination of the above two techniques renders optimal convergence for all $α.$ Numerical results further confirm this conclusion.