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Fast discrete Helmholtz-Hodge decompositions in bounded domains

hal.structure.identifieranalyse appliquée
dc.contributor.authorANGOT, Philippe
hal.structure.identifierTransferts, écoulements, fluides, énergétique [TREFLE]
dc.contributor.authorCALTAGIRONE, Jean-Paul
hal.structure.identifierÉquipe EDP et Physique Mathématique
dc.contributor.authorFABRIE, Pierre
dc.date.accessioned2021-05-14T10:04:18Z
dc.date.available2021-05-14T10:04:18Z
dc.date.created2012-01-31
dc.date.issued2013-04
dc.identifier.issn0893-9659
dc.identifier.urihttps://oskar-bordeaux.fr/handle/20.500.12278/78471
dc.description.abstractEnWe present new fast {\em discrete Helmholtz-Hodge decomposition (DHHD)} methods to efficiently compute at the order $\cO(\eps)$ the divergen\-ce-free (solenoidal) or curl-free (irrotational) components and their associated potentials of a given $\mathbf{L}^2(\Omega)$ vector field in a bounded domain. The solution algorithms solve suitable penalized boundary-value elliptic problems involving either the $\Grad(\Div)$ operator in the {\em vector penalty-projection (VPP)} or the $\Rot(\Rot)$ operator in the {\em rotational penalty-projection (RPP)} with {\em adapted right-hand sides} of the same form. Therefore, they are extremely well-conditioned, fast and cheap avoiding to solve the usual Poisson problems for the scalar or vector potentials. Indeed, each (VPP) or (RPP) problem only requires two conjugate-gradient iterations whatever the mesh size, when the penalty parameter $\varepsilon$ is sufficiently small. We state optimal error estimates vanishing as $\mathcal{O}(\varepsilon)$ with a penalty parameter $\varepsilon$ as small as desired up to machine precision, e.g. $\varepsilon=10^{-14}$. Some numerical results confirm the efficiency of the proposed (DHHD) methods, very useful to solve problems in electromagnetism or fluid dynamics.
dc.language.isoen
dc.publisherElsevier
dc.subject.enHelmholtz-Hodge decompositions
dc.subject.enRotational penalty-projection
dc.subject.enVector penalty-projection
dc.subject.enPenalty method
dc.subject.enError analysis
dc.subject.enPDE's with adapted right-hand sides
dc.title.enFast discrete Helmholtz-Hodge decompositions in bounded domains
dc.typeArticle de revue
dc.identifier.doi10.1016/j.aml.2012.11.006
dc.subject.halMathématiques [math]/Equations aux dérivées partielles [math.AP]
dc.subject.halMathématiques [math]/Analyse numérique [math.NA]
dc.subject.halPhysique [physics]/Physique [physics]/Dynamique des Fluides [physics.flu-dyn]
dc.subject.halPhysique [physics]/Physique [physics]/Physique Numérique [physics.comp-ph]
dc.subject.halSciences de l'ingénieur [physics]/Electromagnétisme
dc.subject.halSciences de l'ingénieur [physics]/Mécanique [physics.med-ph]/Mécanique des fluides [physics.class-ph]
dc.subject.halPhysique [physics]/Mécanique [physics]/Mécanique des fluides [physics.class-ph]
bordeaux.journalApplied Mathematics Letters
bordeaux.page445--451
bordeaux.volume26
bordeaux.hal.laboratoriesInstitut de Mécanique et d’Ingénierie de Bordeaux (I2M) - UMR 5295*
bordeaux.issue4
bordeaux.institutionUniversité de Bordeaux
bordeaux.institutionBordeaux INP
bordeaux.institutionCNRS
bordeaux.institutionINRAE
bordeaux.institutionArts et Métiers
bordeaux.peerReviewedoui
hal.identifierhal-00756959
hal.version1
hal.origin.linkhttps://hal.archives-ouvertes.fr//hal-00756959v1
bordeaux.COinSctx_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:journal&rft.jtitle=Applied%20Mathematics%20Letters&rft.date=2013-04&rft.volume=26&rft.issue=4&rft.spage=445--451&rft.epage=445--451&rft.eissn=0893-9659&rft.issn=0893-9659&rft.au=ANGOT,%20Philippe&CALTAGIRONE,%20Jean-Paul&FABRIE,%20Pierre&rft.genre=article


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