<abstract xml:lang="en" xmlns="http://www.w3.org/1999/xhtml"><p>Patch substructuring methods are non-overlapping domain decomposition methods like classical substructuring methods, but they use information from geometric patches reaching into neighboring subdomains, condensated on the interfaces, to enhance the performance of the method, while keeping it non-overlapping. These methods are very convenient to use in practice, but their convergence properties have not been studied yet. We analyze geometric patch substructuring methods for the special case of one patch per interface. We show that this method is equivalent to an overlapping Schwarz method using Neumann transmission conditions. This equivalence is obtained by first studying a new, algebraic patch method, which is equivalent to the classical Schwarz method with Dirichlet transmission conditions and an overlap corresponding to the size of the patches. Our results motivate a new method, the Robin patch method, which is a linear combination of the algebraic and the geometric one, and can be interpreted as an optimized Schwarz method with Robin transmission conditions. This new method has a significantly faster convergence rate than both the algebraic and the geometric one. We complement our results by numerical experiments.</p></abstract>

Patch substructuring methods are non-overlapping domain decomposition methods like classical substructuring methods, but they use information from geometric patches reaching into neighboring subdomains, condensated on the interfaces, to enhance the performance of the method, while keeping it non-overlapping. These methods are very convenient to use in practice, but their convergence properties have not been studied yet. We analyze geometric patch substructuring methods for the special case of one patch per interface. We show that this method is equivalent to an overlapping Schwarz method using Neumann transmission conditions. This equivalence is obtained by first studying a new, algebraic patch method, which is equivalent to the classical Schwarz method with Dirichlet transmission conditions and an overlap corresponding to the size of the patches. Our results motivate a new method, the Robin patch method, which is a linear combination of the algebraic and the geometric one, and can be interpreted as an optimized Schwarz method with Robin transmission conditions. This new method has a significantly faster convergence rate than both the algebraic and the geometric one. We complement our results by numerical experiments.

Analysis of Patch Substructuring Methods

Section de Mathématiques, Université de Genève, 1211 Genève, Switzerland

LAGA, Institut Galilée, Université Paris, 13 93430 Villetaneuse, France

Applied Mathematics and Systems Laboratory, Ecole Centrale Paris, 92295 Châtenay-Malabry Cedex, France

High Performance Computing Research Unit, ONERA, 92322 Chatillon, France

International Journal of Applied Mathematics and Computer Science

{"article-title":"Analysis of Patch Substructuring Methods"}

Patch substructuring methods are non-overlapping domain decomposition methods like classical substructuring methods, but they use information from geometric...

Analysis of Patch Substructuring Methods

Online veröffentlicht: 11. Okt. 2007

Seitenbereich: 395 - 402

DOI: https://doi.org/10.2478/v10006-007-0032-1

Schlüsselwörter
Schwarz domain decomposition methods, Schur complement methods, patch substructuring methods, optimized Schwarz methods

This content is open access.

Analysis of Patch Substructuring Methods

Online veröffentlicht: 11. Okt. 2007

Seitenbereich: 395 - 402

DOI: https://doi.org/10.2478/v10006-007-0032-1

SchlüsselwörterSchwarz domain decomposition methods, Schur complement methods, patch substructuring methods, optimized Schwarz methods

This content is open access.

Schlüsselwörter
Schwarz domain decomposition methods, Schur complement methods, patch substructuring methods, optimized Schwarz methods