A Cascadic Multigrid Method for Semilinear Elliptic Equations

Abstract

This paper introduce a cascadic multigrid method for solving semilinear elliptic equations based on a multilevel correction method. Instead of the common costly way of directly solving semilinear elliptic equation on a very fine space, the new method contains some smoothing steps on a series of multilevel finite element spaces and some solving steps to semilinear elliptic equations on a very coarse space. To prove the efficiency of the new method, we derive two results, one of the optimal convergence rate by choosing the appropriate sequence of finite element spaces and the number of smoothing steps, and the other of the optimal computational work by applying the parallel computing technique. Moreover, the requirement of bounded second order derivatives of nonlinear term in the existing multigrid methods is reduced to a bounded first order derivative in the new method. Some numerical experiments are presented to validate our theoretical analysis.