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

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

A Locking-Free Weak Galerkin Finite Element Method for Linear Elasticity Problems

In this paper, we introduce and analyze a lowest-order locking-free weak Galerkin (WG) finite element scheme for the grad-div formulation of linear elasticity problems. The scheme uses linear functions in the interior of mesh elements and constants on edges (2D) or faces (3D), respectively, to approximate the displacement. An $H(div)$-conforming displacement reconstruction operator is employed to modify test functions in the right-hand side of the discrete form, in order to eliminate the dependence of the $Lam\acute{e}$ parameter $\lambda$ in error estimates, i.e., making the scheme locking-free. The method works without requiring $\lambda \|\nabla\cdot \mathbf{u}\|_1$ to be bounded. We prove optimal error estimates, independent of $\lambda$, in both the $H^1$-norm and the $L^2$-norm. Numerical experiments validate that the method is effective and locking-free