Domain decomposition (DD) and multigrid (MG) have been intensively studied for linear elliptic problems. Their convergence can be analysed using the framework of space decomposition and subspace correction. Using this framework, we extend the DD and MG methods to a class of nonlinear problems. We shall first show that our proposed algorithms have a convergence rate which is as good as for linear elliptic problems when the nonlinear operator is strongly monotone and Lipschtz continuous. We shall also give a convergence rate estimate for nonlinear degerated and singular elliptic problems. Applications will be discussed for nonlinear p-Laplace equation and the full potential flow equation. Some numerical results will be reported. Convergence of some asynchronous version will also be discussed.
1 This work was partially supported by the Norwegian Research Council Strategic Institute Programm within Inverse Problems at RF-Rogaland Reseach and by Project SEP-115837/431 at Mathematics Institute, University of Bergen.
2 This work was partially supported by NSF DMS-9706949 through Penn State.