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Convergence of the Dirichlet-Neumann method for semilinear elliptic equations

Emil Engström

Abstract

The Dirichlet-Neumann method is a common domain decomposition method for nonoverlapping domain decomposition and the method has been studied extensively for linear elliptic equations. However, for nonlinear elliptic equations, there are only convergence results for some specific cases in one spatial dimension. The aim of this manuscript is therefore to prove that the Dirichlet-Neumann method converges for a class of semilinear elliptic equations on Lipschitz continuous domains in two and three spatial dimensions. This is achieved by first proving a new result on the convergence of nonlinear iterations in Hilbert spaces and then applying this result to the Steklov-Poincaré formulation of the Dirichlet-Neumann method.

Convergence of the Dirichlet-Neumann method for semilinear elliptic equations

Abstract

The Dirichlet-Neumann method is a common domain decomposition method for nonoverlapping domain decomposition and the method has been studied extensively for linear elliptic equations. However, for nonlinear elliptic equations, there are only convergence results for some specific cases in one spatial dimension. The aim of this manuscript is therefore to prove that the Dirichlet-Neumann method converges for a class of semilinear elliptic equations on Lipschitz continuous domains in two and three spatial dimensions. This is achieved by first proving a new result on the convergence of nonlinear iterations in Hilbert spaces and then applying this result to the Steklov-Poincaré formulation of the Dirichlet-Neumann method.

Paper Structure

This paper contains 6 sections, 6 theorems, 102 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Abstract theorem on nonlinear splitting in Hilbert spaces
  3. Weak formulations of semilinear elliptic equations
  4. Transmission problem and Steklov--Poincaré operators
  5. Convergence of the Dirichlet--Neumann method
  6. Numerical results

Key Result

Theorem 2.1

\newlabelthm:abstract0 Let $X$ be a (real) Hilbert space, $G:X \rightarrow X^*$ be a uniformly monotone operator, $\chi\in X^*$, and $\eta^0\in X$.

Figures (4)

  • Figure 1: The domain and domain decomposition used in our numerical example (left) and the mesh exemplified with $h=1/2$ (right).
  • Figure 2: The error of the Dirichlet--Neumann applied to \ref{['ex:ex1']} (left) and \ref{['ex:ex2']} (right) together with the decomposition as in \ref{['fig:L']}. Three different values of the mesh parameter $h$ is used, but for most of the iterations, they all have the same error and therefore only one can be seen.
  • Figure 3: The error of the Dirichlet--Neumann method (DN) applied to \ref{['ex:ex1']} (left) and \ref{['ex:ex2']} (right) compared to the Neumann--Neumann (NN) and Robin--Robin (RR) methods.
  • Figure :

Theorems & Definitions (14)

  • Theorem 2.1
  • Proof 1
  • Example 3.3
  • Example 3.4
  • Theorem 3.5
  • Proof 2
  • Theorem 4.1
  • Proof 3
  • Theorem 4.2
  • Proof 4
  • ...and 4 more