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A meshfree method with domain decomposition for Helmholtz boundary value problems

2020-11Conference paper Open access
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datacite.subject.fosCiências Naturais::Matemáticas
dc.contributor.authorValtchev, Svilen
dc.date.accessioned2025-07-11T17:06:19Z
dc.date.available2025-07-11T17:06:19Z
dc.date.issued2020-11
dc.descriptionEISBN - 978-1-7281-8840-9
dc.descriptionArticle number - 9280636; Conference name - 2nd International Conference on Control Systems, Mathematical Modeling, Automation and Energy Efficiency, SUMMA 2020; Conference city - Lipetsk; Conference date - 10 November 2020 - 13 November 2020; Conference code - 165770
dc.description.abstractIn the framework of meshfree methods, we address the numerical solution of boundary value problems (BVP) for the non-homogeneous modified Helmholtz partial differential equation (PDE). In particular, the unknown solution of the BVP is calculated in two steps. First, a particular solution of the PDE is approximated by superposition of plane wave functions with different wavenumbers and directions of propagation. Then, the corresponding homogeneous BVP is solved, for the homogeneous part of the solution, using the classical method of fundamental solutions (MFS). The combination of these two meshfree techniques shows excellent numerical results for non-homogeneous BVPs posed in simple geometries and when the source term of the PDE is sufficiently regular. However, for more complex domains or when the source term is piecewise defined, the MFS fails to converge. We overcome this problem by coupling the MFS with Lions non-overlapping domain decomposition method. The proposed technique is tested for the modified Helmholtz PDE with a discontinuous source term, posed in an L-shaped domain.eng
dc.description.sponsorshipThe financial support from the Portuguese FCT - Fundação para a Ciência e a Tecnologia, through the projects UIDB/04621/2020 and UIDP/04621/2020 of CEMAT/IST-ID, Center for Computational and Stochastic Mathematics, Instituto Superior Técnico, University of Lisbon, is gratefully acknowledged.
dc.identifier.citationS. S. Valtchev, "A meshfree method with domain decomposition for Helmholtz boundary value problems," 2020 2nd International Conference on Control Systems, Mathematical Modeling, Automation and Energy Efficiency (SUMMA), Lipetsk, Russia, 2020, pp. 139-144, doi: https://doi.org/10.1109/SUMMA50634.2020.9280636.
dc.identifier.doi10.1109/summa50634.2020.9280636
dc.identifier.isbn978-1-7281-8113-4
dc.identifier.isbn978-1-7281-8840-9
dc.identifier.urihttp://hdl.handle.net/10400.8/13613
dc.language.isoeng
dc.peerreviewedyes
dc.publisherIEEE Canada
dc.relationCenter for Computational and Stochastic Mathematics
dc.relation.hasversionhttps://ieeexplore.ieee.org/document/9280636
dc.relation.ispartof2020 2nd International Conference on Control Systems, Mathematical Modeling, Automation and Energy Efficiency (SUMMA)
dc.rights.uriN/A
dc.subjectmeshfree method
dc.subjectplane wave functions
dc.subjectmethod of fundamental solutions
dc.subjectdomain decomposition
dc.subjectmodified Helmholtz equation
dc.subjectnon-homogeneous PDE
dc.subjectL-shaped domain
dc.titleA meshfree method with domain decomposition for Helmholtz boundary value problemseng
dc.typeconference paper
dspace.entity.typePublication
oaire.awardTitleCenter for Computational and Stochastic Mathematics
oaire.awardURIhttp://hdl.handle.net/10400.8/13612
oaire.citation.conferenceDate2020-11
oaire.citation.conferencePlaceLipetsk, Russia
oaire.citation.endPage144
oaire.citation.startPage139
oaire.citation.title2020 2nd International Conference on Control Systems, Mathematical Modeling, Automation and Energy Efficiency (SUMMA)
oaire.fundingStream6817 - DCRRNI ID
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
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relation.isProjectOfPublication.latestForDiscovery9c9950fd-9a9f-4c8f-94eb-65df530d33e9

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In the framework of meshfree methods, we address the numerical solution of boundary value problems (BVP) for the non-homogeneous modified Helmholtz partial differential equation (PDE). In particular, the unknown solution of the BVP is calculated in two steps. First, a particular solution of the PDE is approximated by superposition of plane wave functions with different wavenumbers and directions of propagation. Then, the corresponding homogeneous BVP is solved, for the homogeneous part of the solution, using the classical method of fundamental solutions (MFS). The combination of these two meshfree techniques shows excellent numerical results for non-homogeneous BVPs posed in simple geometries and when the source term of the PDE is sufficiently regular. However, for more complex domains or when the source term is piecewise defined, the MFS fails to converge. We overcome this problem by coupling the MFS with Lions non-overlapping domain decomposition method. The proposed technique is tested for the modified Helmholtz PDE with a discontinuous source term, posed in an L-shaped domain.
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