Convergence of adaptive two-grid weak Galerkin finite element methods for semilinear elliptic differential equations

Abstract

In this paper, we investigate the convergence of an adaptive two-grid weak Galerkin (ATGWG) finite element method for second order semilinear elliptic partial differential equations (PDEs). First, we propose an ATGWG method and then prove that the sum of the energy error and the error estimator of ATGWG method between two consecutive adaptive loops is a contraction. The weak Galerkin (WG) elements (Pj(T),Pℓ(∂T),RTj(T)) (Wang and Ye, 2013) are studied in this paper and numerical experiments based on the lowest order case with j=l=0 are provided to support the theoretical results.