Contemporary Mathematics Volume Domain Decomposition Methods for Monotone Nonlinear Elliptic Problems

In this paper we study several overlapping domain decompo sition based iterative algorithms for the numerical solution of some non linear strongly elliptic equations discretized by the nite element methods In particular we consider additive Schwarz algorithms used together with the classical inexact Newton methods We show that the algorithms con verge and the convergence rates are independent of the nite element mesh parameter as well as the number of subdomains used in the domain de composition Introduction Schwarz type overlapping domain decomposition methods have been studied extensively in the past few years for linear elliptic nite element problems see e g In this paper we extend some of the theory and methods to the class of nonlinear strongly elliptic nite element problems The rst study of the classical Schwarz alternating method for nonlinear elliptic equations appeared in the paper of P L Lions in which the class of continuous monotonic elliptic problems was investigated There are basically two approaches that a domain decomposition method can be used to solve a nonlinear problem The rst approach is to locally linearize the nonlinear equation via a Newton like algorithm and then to solve the resulting linearized problems at each nonlinear iteration by a domain decomposition method The second approach is to use domain decomposition such as the Schwarz alternating method directly on the Mathematics Subject Classi cation F N N