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A spectral element solution of the Poisson equation with shifted boundary polynomial corrections: influence of the surrogate to true boundary mapping and an asymptotically preserving Robin formulation

hal.structure.identifierDepartment of Applied Mathematics and Computer Science [Lyngby] [DTU Compute]
dc.contributor.authorVISBECH, Jens
hal.structure.identifierDepartment of Applied Mathematics and Computer Science [Lyngby] [DTU Compute]
dc.contributor.authorENGSIG-KARUP, Allan Peter
hal.structure.identifierCertified Adaptive discRete moDels for robust simulAtions of CoMplex flOws with Moving fronts [CARDAMOM]
dc.contributor.authorRICCHIUTO, Mario
dc.date.accessioned2024-04-04T02:32:01Z
dc.date.available2024-04-04T02:32:01Z
dc.date.issued2023
dc.identifier.urihttps://oskar-bordeaux.fr/handle/20.500.12278/190348
dc.description.abstractEnWe present a new high-order accurate spectral element solution to the two-dimensional scalar Poisson equation subject to a general Robin boundary condition. The solution is based on a simplified version of the shifted boundary method employing a continuous arbitrary order $hp$-Galerkin spectral element method as the numerical discretization procedure. The simplification relies on a polynomial correction to avoid explicitly evaluating high-order partial derivatives from the Taylor series expansion, which traditionally have been used within the shifted boundary method. In this setting, we apply an extrapolation and novel interpolation approach to project the basis functions from the true domain onto the approximate surrogate domain. The resulting solution provides a method that naturally incorporates curved geometrical features of the domain, overcomes complex and cumbersome mesh generation, and avoids problems with small-cut-cells. Dirichlet, Neumann, and general Robin boundary conditions are enforced weakly through: i) a generalized Nitsche's method and ii) a generalized Aubin's method. For this, a consistent asymptotic preserving formulation of the embedded Robin formulations is presented. We present several numerical experiments and analysis of the algorithmic properties of the different weak formulations. With this, we include convergence studies under polynomial, $p$, increase of the basis functions, mesh, $h$, refinement, and matrix conditioning to highlight the spectral and algebraic convergence features, respectively. This is done to assess the influence of errors across variational formulations, polynomial order, mesh size, and mappings between the true and surrogate boundaries.
dc.language.isoen
dc.rights.urihttp://creativecommons.org/licenses/by/
dc.subject.enNumerical Analysis (math.NA)
dc.subject.enFOS: Mathematics
dc.subject.enSpectral element method
dc.subject.enshifted boundary method
dc.subject.enhigh-order numerical method
dc.subject.enembedded methods
dc.subject.enPoisson problem
dc.subject.enelliptic problem
dc.subject.enDirichlet
dc.subject.enNeumann
dc.subject.enand Robin boundary conditions
dc.title.enA spectral element solution of the Poisson equation with shifted boundary polynomial corrections: influence of the surrogate to true boundary mapping and an asymptotically preserving Robin formulation
dc.typeDocument de travail - Pré-publication
dc.identifier.doi10.48550/arXiv.2310.17621
dc.subject.halMathématiques [math]
bordeaux.hal.laboratoriesInstitut de Mathématiques de Bordeaux (IMB) - UMR 5251*
bordeaux.institutionUniversité de Bordeaux
bordeaux.institutionBordeaux INP
bordeaux.institutionCNRS
hal.identifierhal-04342005
hal.version1
hal.origin.linkhttps://hal.archives-ouvertes.fr//hal-04342005v1
bordeaux.COinSctx_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:journal&rft.date=2023&rft.au=VISBECH,%20Jens&ENGSIG-KARUP,%20Allan%20Peter&RICCHIUTO,%20Mario&rft.genre=preprint


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