An Over-Penalized Weak Galerkin Method for Second-Order Elliptic Problems

Abstract

The weak Galerkin (WG) finite element method was first introduced by Wang and Ye for solving second order elliptic equations, with the use of weak functions and their weak gradients. The basis function spaces depend on different combinations of polynomial spaces in the interior subdomains and edges of elements, which makes the WG methods flexible and robust in many applications. Different from the definition of jump in discontinuous Galerkin (DG) methods, we can define a new weaker jump from weak functions defined on edges. Those functions have double values on the interior edges shared by two elements rather than a limit of functions defined in an element tending to its edge. Naturally, the weak jump comes from the difference between two weak functions defined on the same edge. We introduce an over-penalized weak Galerkin (OPWG) method, which has two sets of edge-wise and element-wise shape functions, and adds a penalty term to control weak jumps on the interior edges. Furthermore, optimal a priori error estimates in $H^1$ and $L^2$ norms are established for the finite element ($\mathbb{P}_k(K)$, $\mathbb{P}_k(e)$, $RT_k(K)$). In addition, some numerical experiments are given to validate theoretical results, and an incomplete LU decomposition has been used as a preconditioner to reduce iterations from the GMRES, CG, and BICGSTAB iterative methods.