A two-level overlapping Schwarz preconditioner for discontinuous Galerkin methods

Calvo, Juan G.; Solano, Moisés; Calvo, Juan G.; Solano, Moisés

doi:10.15517/rmta.v32i1.59472

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Revista de Matemática Teoría y Aplicaciones

Print version ISSN 1409-2433

Rev. Mat vol.32 n.1 San José Jan./Mar. 2025

http://dx.doi.org/10.15517/rmta.v32i1.59472

Artículo

A two-level overlapping Schwarz preconditioner for discontinuous Galerkin methods

Un precondicionador de Schwarz de dos niveles con traslape para métodos discontinuos de Galerkin

Juan G. Calvo¹

Moisés Solano²

^¹Universidad de Costa Rica, Centro de Investigación en Matemática Pura y Aplicada, San José, Costa Rica; juan.calvo@ucr.ac.cr

^²Universidad de Costa Rica, Escuela de Matemática, San José, Costa Rica; moises.solano@ucr.acr.cr

Abstract

This article introduces a two-level overlapping additive Schwarz algorithm tailored for solving elliptic problems discretized with the symmetric interior penalty discontinuous Galerkin method. The proposed algorithm allows for the use of irregular subdomains, overcoming limitations of other approaches where the coarse mesh was based on triangular elements. Additionally, we provide a brief description of the numerical implementation of the Galerkin method. We present numerical results validating the relevance of our algorithm, including cases where the coefficient of the differential equation is discontinuous-a feature that is particularly relevant to various practical applications.

Keywords: Domain decomposition; Discontinuous Galerkin methods; Irregular subdomain boundaries; Overlapping Schwarz algorithms; Nodal elliptic problems.

Resumen

Este artículo presenta un algoritmo aditivo de Schwarz de dos niveles con traslape diseñado para resolver problemas elípticos discretizados con el método Galerkin discontinuo de penalización interior simétrico. El algoritmo propuesto permite utilizar subdominios irregulares, superando limitaciones de otros enfoques donde la malla gruesa se basaba en elementos triangulares. Se incluye además una breve descripción de la implementación numérica del método de Galerkin. Se presentan resultados numéricos que validan la pertinencia del método, incluyendo casos donde el coeficiente de la ecuación diferencial es discontinuo, una característica que es relevante en diversas aplicaciones.

Palabras clave: Descomposición de dominios; Métodos discontinuos de Galerkin; Subdominios con frontera irregular; Algoritmos con traslape de Schwarz; Problemas elípticos nodales.

Mathematics Subject Classification: Primary: 65F08, 65F10. Secondary: 65N30, 65N55.

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Acknowledgments

Authors gratefully acknowledge the institutional support for Project 821-C1-228 subscribed to the Vice-Rectory for Research, University of Costa Rica.

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Received: June 05, 2024; Accepted: December 17, 2024

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