@Article{CiCP-32-1094,
author = {Liu , YiChen , Wenbin and Wang , Yanqiu},
title = {A Weak Galerkin Mixed Finite Element Method for Second Order Elliptic Equations on 2D Curved Domains},
journal = {Communications in Computational Physics},
year = {2022},
volume = {32},
number = {4},
pages = {1094--1128},
abstract = {<p style="text-align: justify;">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 $α&gt;1.$ 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 $α ≥ 1.$ It is a little
surprising to see that even the lowest-order WG-MFEM $(α = 1)$ 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 $α.$ Numerical results further confirm this conclusion.</p>},
issn = {1991-7120},
doi = {https://doi.org/10.4208/cicp.OA-2022-0106},
url = {http://global-sci.org/intro/article_detail/cicp/21140.html}
}