A Cascadic Algorithm for the Solution of Nonlinear Sign-indefinite Problems

Abstract

. In this paper we propose a cascadic algorithm for a weakly nonlinear sign-indefinite elliptic Dirichlet problem. The standard finite element discretization with piecewise linear finite elements leads to a system of nonlinear equations. We solve these nonlinear equations by a cascadic organization of Newton's method with "frozen derivatives" on a sequence of nested grids. This gives a simple version of a multigrid method without projections on coarser grids. The cascadic algorithm starts on a comparatively coarse grid where the number of unknowns is small enough to obtain an approximate solution within sufficiently high precision without substantial computational effort. On each finer grid we perform exactly one Newton step taking the approximate solution from the coarsest grid as initial guess. The arising linear systems are solved iteratively by Jacobi-type iterations with special parameters. As initial guess for the iterations we use the approximate solution from the previous grid...