A Cascadic Algorithm for the Solution Ofnonlinear Sign - Indefinite Problemsvladimir

In this paper we propose a cascadic algorithm for a weakly non-linear sign-indeenite elliptic Dirichlet problem. The standard nite element discretization with piecewise linear nite elements leads to a system of non-linear 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 suuciently high precision without substantial computational eeort. On each ner 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. We prove that for suuciently ne initial grid and for a suuciently good start approximation the algorithm is optimal with respect to accuracy and computational complexity. 1. Introduction During the last years a special kind of \one-way multigrid" method was developed which uses a sequence of successively ner grids without projection on coarser meshes. First, P. Deuuhard presented a cascadic conjugate gradient method, Deu93], where the high speed of convergence was demonstrated computational-ly. In the following years optimal arithmetic complexity was proved for second order elliptic boundary value problems with smooth solutions, Sha93], and H 1+-regular problems, Sha94], Bor94], BD96]. Moreover, the convergence analysis was extended to a whole class of smoothing iterations, Bor94], and to problems with curvilinear boundary, Sha95b]. In all cited papers the second order elliptic operator was linear, self-adjoint and positive-deenite. Recently, L. V. Gilyova and V. V. Shaidurov proposed a cas-cadic algorithm for a nonlinearly perturbed elliptic Dirichlet problem, GS98a].