Two‐grid weak Galerkin method for semilinear elliptic differential equations

Abstract

In this paper, we investigate a two‐grid weak Galerkin method for semilinear elliptic differential equations. The method mainly contains two steps. First, we solve the semilinear elliptic equation on the coarse mesh with mesh size , then, we use the coarse mesh solution as an initial guess to linearize the semilinear equation on the fine mesh, that is, on the fine mesh (with mesh size ), we only need to solve a linearized system. Theoretical analysis shows that when the exact solution has sufficient regularity and , the two‐grid weak Galerkin method achieves the same convergence accuracy as weak Galerkin method. Several examples are given to verify the theoretical results.