A Cascadic Algorithm for Nonlinearly Perturbed Elliptic Dirichlet Problems

In this paper we propose a cascadic algorithm for nonlinearly perturbed elliptic Dirichlet problems. The nonlinear equations arising from certain nite element discretizations are solved by Newton's method. The cascadic algorithm starts on a comparatively coarse grid where the number of unknowns is small enough to obtain an approximate solution within suu-ciently high precision without substantial computational eeort. On each grid we perform exactly one Newton step taking the approximate solution from the previous grid as initial guess. The linear systems are solved iteratively by an appropriate smoothing method, in particular by the conjugate gradient method. We prove that the algorithm has optimal complexity and that for suuciently good start approximations it converges to an approximate solution the error of which is of the same order as the discretization error. 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 going back to coarser meshes. This type of cascadic algorithm was rst introduced by P. Deuu-hard in Deu93]. There he proposed a cascadic conjugate gradient method and demonstrated its high speed of convergence by numerical examples. In Sha93], Sha96] V. V. Shaidurov proved optimal complexity for this algorithm in case of H 2-regular second order elliptic equations. In Sha94], Bor94], BD96] the optimal complexity was shown for two-and three-dimensional H 1+-regular problems. Moreover, in Bor94], BD96] F. A. Bornemann and P. Deuuhard extended the convergence analysis to a whole class of smoothing iterations. In Sha95b] V. V. Shaidurov presented a proof for domains with curvilinear boundary. In all cited papers only linear boundary value problems were considered. Recently , L. V. Gilyova and V. V. Shaidurov proposed a cascadic algorithm for a two-dimensional weakly nonlinear elliptic Dirichlet problem, see GS97]. They used Newton's method with frozen derivatives, i.e. with a xed Jacobian evaluated at the coarsest grid, and conjugate gradient or a special Jacobi-type iteration for