Cross-points in the Dirichlet-Neumann method II: a geometrically convergent variant

TL;DR

This paper introduces a new variant of the Dirichlet-Neumann method that ensures geometric convergence even with cross-points in domain decompositions, supported by theoretical proofs and numerical experiments.

Contribution

A novel Dirichlet-Neumann method variant that maintains well-posedness and geometric convergence in the presence of cross-points, extending previous results.

Findings

Proved geometric convergence of the new method with cross-points.

Extended convergence results from 2D to 3D cases.

Numerical experiments confirm theoretical predictions.

Abstract

When considered as a standalone iterative solver for elliptic boundary value problems, the Dirichlet-Neumann (DN) method is known to converge geometrically for domain decompositions into strips, even for a large number of subdomains. However, whenever the domain decomposition includes cross-points, i.e.$\!$ points where more than two subdomains meet, the convergence proof does not hold anymore as the method generates subproblems that might not be well-posed. Focusing on a simple two-dimensional example involving one cross-point, we proposed in a previous work a decomposition of the solution into two parts: an even symmetric part and an odd symmetric part. Based on this decomposition, we proved that the DN method was geometrically convergent for the even symmetric part and that it was not well-posed for the odd symmetric part. Here, we introduce a new variant of the DN method which generates subproblems that remain well-posed for the odd symmetric part as well. Taking advantage of the symmetry properties of the domain decomposition considered, we manage to prove that our new method converges geometrically in the presence of cross-points. We also extend our results to the three-dimensional case, and present numerical experiments that illustrate our theoretical findings.