A Two-Grid Finite Element Method for Nonlinear Sobolev Equations

Abstract

A two-grid based finite element method for nonlinear Sobolev equations is studied. It consists in solving small nonlinear systems related to coarse-grids, following the solution of linear systems in fine-grid spaces. The method has the same accuracy as the standard finite element method but reduces workload and saves CPU time. The $H^1$-error estimates show that the two-grid methods have optimal convergence if the coarse $H$ and fine $h$ mesh sizes satisfy the condition $h=\mathscr{O}(H^2)$. Numerical examples confirm the theoretical findings.