On the convergence of the nonlinear multigrid algorithm for potential operators

Abstract

Among numerical methods to solve large-scale algebraic equations systems (which may result from the approximation of two-or three-dimensional integral or differential equations), the multi-grid-methods have been rapidly developed during the last 10 years. These methods have the property that the necessary effort is proportional to the number of unknown parameters. In the present paper we give results on a non-linear multigrid method that supports an efficient solution of non-linear problems without linearization. Thereby the linear convergence of the method can be shown under rather general assumptions. For the proofs we use mappings with potentials (cf. Daniel [1]). Under stronger assumptions we show for a two-level-algorithm the rate of convergence to be independent of the level of refinements. Our results continue the investigation done by Hackbusch and Reusken [2–5].