Iterative two-grid methods for semilinear elliptic equations

Abstract

We design and analyze a new finite element discretization technique based on iterative two-grid methods for semilinear elliptic partial differential equations (PDEs). We first present an iterative two-grid method that just applies the classical two-grid for semilinear elliptic PDEs in a successive fashion, namely, to solve a semilinear problem on the coarse space and then to solve a symmetric positive definite problem on the fine space. Thus, we only need to deal with semilinear problems in the coarse space in the first algorithm. Secondly, we replace the semilinear term by using the corresponding first order Taylor expansion in the second iterative two-grid. We provide the error estimates for the second iterative algorithm and present numerical experiments to show the efficiency of our methods.