A nonlinear adaptative multiresolution method in finite differences with incremental unknowns

Abstract

In this article, we propose a new method well suited for the calculation of unstable solutions of nonlinear eigenvalues problem. This method is derived from the classical Marder-Weitzner scheme (MW) which can be seen as a nonlinear Richardson method. First we adapt to (MW) the usual extension of the classical Linear Richardson scheme (LR) which consists in computing the relaxation parameter in order to minimize the iterative residual in a suitable norm. This method is then generalized with the utilization of the Incremental Unknowns (I.U.) inducing the minimizing relaxation parameter in the embedded hierarchical subspaces. We obtain in this way both generalizations of the MW and the LR algorithms. The numerical illustrations we give allowing comparisons between the differents LR schemes (for linear problems) and some versions of the MW method (for nonlinear eigenvalue problems), point out the better speed of convergence of the new algorithms.