On the nonlinear Dirichlet-Neumann method and preconditioner for Newton's method

TL;DR

This paper analyzes the nonlinear Dirichlet-Neumann method as an iterative solver and preconditioner for Newton's method, demonstrating conditions for quadratic convergence and mesh-independent performance.

Contribution

It introduces a theoretical framework for the nonlinear DN method, proving convergence properties and its effectiveness as a preconditioner for Newton's method.

Findings

Existence of a relaxation parameter for quadratic convergence.

Convergence of preconditioned Newton's method is independent of relaxation.

Numerical results show mesh-independent convergence and competitive performance.

Abstract

The Dirichlet-Neumann (DN) method has been extensively studied for linear partial differential equations, while little attention has been devoted to the nonlinear case. In this paper, we analyze the DN method both as a nonlinear iterative method and as a preconditioner for Newton's method. We discuss the nilpotent property and prove that under special conditions, there exists a relaxation parameter such that the DN method converges quadratically. We further prove that the convergence of Newton's method preconditioned by the DN method is independent of the relaxation parameter. Our numerical experiments further illustrate the mesh independent convergence of the DN method and compare it with other standard nonlinear preconditioners.