  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.
