A Systematic Study on Weak Galerkin Finite Element Methods for Second Order Elliptic Problems

Abstract

This article provides a systematic study for the weak Galerkin (WG) finite element method for second order elliptic problems by exploring polynomial approximations with various degrees for each local element. A typical local WG element is of the form $$P_k(T)\times P_j(\partial T)\Vert P_\ell (T)^2$$Pk(T)?Pj(źT)źPl(T)2, where $$k\ge 1$$kź1 is the degree of polynomials in the interior of the element T, $$j\ge 0$$jź0 is the degree of polynomials on the boundary of T, and $$\ell \ge 0$$lź0 is the degree of polynomials employed in the computation of weak gradients or weak first order partial derivatives. A general framework of stability and error estimate is developed for the corresponding numerical solutions. Numerical results are presented to confirm the theoretical results. The work reveals some previously undiscovered strengths of the WG method for second order elliptic problems, and the results are expected to be generalizable to other type of partial differential equations.