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Domain decomposition methods with fundamental solutions for Helmholtz problems with discontinuous source terms

2021-04-15Artigo científico Acesso restrito
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datacite.subject.fosCiências Naturais::Matemáticas
datacite.subject.fosCiências Naturais::Ciências da Computação e da Informação
datacite.subject.sdg03:Saúde de Qualidade
datacite.subject.sdg07:Energias Renováveis e Acessíveis
datacite.subject.sdg12:Produção e Consumo Sustentáveis
dc.contributor.authorAlves, Carlos J. S.
dc.contributor.authorMartins, Nuno F. M.
dc.contributor.authorValtchev, Svilen S.
dc.date.accessioned2026-03-10T12:57:22Z
dc.date.available2026-03-10T12:57:22Z
dc.date.issued2021-04-15
dc.description.abstractThe direct application of the classical method of fundamental solutions (MFS) is restricted to homogeneous linear partial differential equations (PDEs). The use of fundamental solutions with different frequencies allowed the extension of the MFS to non-homogeneous PDEs, in particular, for Poisson or Helmholtz equations and for elastostatic or elastodynamic problems. This method has been called method of fundamental solutions for domains (MFS-D), but it faces an approximation problem when the non-homogeneous term presents discontinuities, because the fundamental solutions are analytic functions outside the source point set. In this paper we analyze two domain decomposition techniques for overcoming this approximation problem. The problem is set in the context of the modified Helmholtz equation, and we also establish the missing density results that justify both the MFS and the MFS-D approximations. Numerical results are presented comparing a direct and an iterative domain decomposition technique, with simulations in non-trivial domains.eng
dc.description.sponsorshipThe financial support from CEMAT, Portugal, CMA, Portugal and Fundação para a Ciência e a Tecnologia (FCT), Portugal , through the projects UID/Multi/04621/2013 (first and third author) and EXCL/MAT-NAN/0114/2012 (first and second author) is gratefully acknowledged.
dc.identifier.citationCarlos J.S. Alves, Nuno F.M. Martins, Svilen S. Valtchev, Domain decomposition methods with fundamental solutions for Helmholtz problems with discontinuous source terms, Computers & Mathematics with Applications, Volume 88, 2021, Pages 16-32, ISSN 0898-1221, https://doi.org/10.1016/j.camwa.2018.12.014.
dc.identifier.doi10.1016/j.camwa.2018.12.014
dc.identifier.issn0898-1221
dc.identifier.issn1873-7668
dc.identifier.urihttp://hdl.handle.net/10400.8/15822
dc.language.isoeng
dc.peerreviewedyes
dc.publisherElsevier
dc.relationCenter for Computational and Stochastic Mathematics
dc.relation.hasversionhttps://www.sciencedirect.com/science/article/pii/S0898122118307065?via%3Dihub
dc.relation.ispartofComputers & Mathematics with Applications
dc.rights.urihttp://creativecommons.org/licenses/by/4.0/
dc.subjectModified Helmholtz equation
dc.subjectNon-homogeneous problems
dc.subjectDomain decomposition
dc.subjectMethod of fundamental solutions
dc.subjectDensity results
dc.titleDomain decomposition methods with fundamental solutions for Helmholtz problems with discontinuous source termseng
dc.typejournal article
dspace.entity.typePublication
oaire.awardNumberUID/Multi/04621/2013
oaire.awardTitleCenter for Computational and Stochastic Mathematics
oaire.awardURIhttp://hdl.handle.net/10400.8/15821
oaire.citation.endPage32
oaire.citation.startPage16
oaire.citation.titleComputers & Mathematics with Applications
oaire.citation.volume88
oaire.fundingStreamFinanciamento do Plano Estratégico de Unidades de I&D - 2013/2015 - OE
oaire.versionhttp://purl.org/coar/version/c_970fb48d4fbd8a85
person.familyNameValtchev
person.givenNameSvilen
person.identifier.ciencia-idAF1E-BD9D-A8D7
person.identifier.gsidhttps://scholar.google.com/citations?user=MtxwhiUAAAAJ&hl=en
person.identifier.orcid0000-0002-3474-2788
person.identifier.scopus-author-id8361079200
relation.isAuthorOfPublicationb6302c21-a0e4-4419-967b-0a1bac949132
relation.isAuthorOfPublication.latestForDiscoveryb6302c21-a0e4-4419-967b-0a1bac949132
relation.isProjectOfPublicationd8281575-89e3-4b28-bbb6-ceb67f204e05
relation.isProjectOfPublication.latestForDiscoveryd8281575-89e3-4b28-bbb6-ceb67f204e05

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The direct application of the classical method of fundamental solutions (MFS) is restricted to homogeneous linear partial differential equations (PDEs). The use of fundamental solutions with different frequencies allowed the extension of the MFS to non-homogeneous PDEs, in particular, for Poisson or Helmholtz equations and for elastostatic or elastodynamic problems. This method has been called method of fundamental solutions for domains (MFS-D), but it faces an approximation problem when the non-homogeneous term presents discontinuities, because the fundamental solutions are analytic functions outside the source point set. In this paper we analyze two domain decomposition techniques for overcoming this approximation problem. The problem is set in the context of the modified Helmholtz equation, and we also establish the missing density results that justify both the MFS and the MFS-D approximations. Numerical results are presented comparing a direct and an iterative domain decomposition technique, with simulations in non-trivial domains.
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