Stability estimates for h-p spectral element methods for general elliptic problems on curvilinear domains

Pravir Dutt, Satyendra Tomar

    Research output: Contribution to journalArticleAcademicpeer-review

    13 Citations (Scopus)

    Abstract

    In this paper we show that the h-p spectral element method developed in 3,8,9 applies to elliptic problems in curvilinear polygons with mixed Neumann and Dirichlet boundary conditions provided that the Babuska–Brezzi inf-sup conditions are satisfied. We establish basic stability estimates for a non-conforming h-p spectral element method which allows for simultaneous mesh refinement and variable polynomial degree. The spectral element functions are non-conforming if the boundary conditions are Dirichlet. For problems with mixed boundary conditions they are continuous only at the vertices of the elements. We obtain a stability estimate when the spectral element functions vanish at the vertices of the elements, which is needed for parallelizing the numerical scheme. Finally, we indicate how the mesh refinement strategy and choice of polynomial degree depends on the regularity of the coefficients of the differential operator, smoothness of the sides of the polygon and the regularity of the data to obtain the maximum accuracy achievable.
    Original languageEnglish
    Pages (from-to)395-429
    Number of pages35
    JournalProceedings of the Indian academy of sciences - mathematical sciences
    Volume113
    Issue number4
    DOIs
    Publication statusPublished - Nov 2003

    Fingerprint

    Spectral Element Method
    Spectral Elements
    Stability Estimates
    Mesh Refinement
    Mixed Boundary Conditions
    Elliptic Problems
    Polygon
    Regularity
    Inf-sup Condition
    Polynomial
    Neumann Boundary Conditions
    Numerical Scheme
    Dirichlet Boundary Conditions
    Dirichlet
    Differential operator
    Vanish
    Smoothness
    Boundary conditions
    Coefficient
    Strategy

    Keywords

    • IR-70531
    • curvilinear polygons
    • geometrical mesh
    • METIS-213783
    • mixed Neumann Dirichlet boundary conditions
    • fractional Sobolev norms
    • infsup conditions
    • stability estimates
    • Corner singularities - geometrical mesh - mixed Neumann and Dirichlet boundary conditions - curvilinear polygons - inf-sup conditions - stability estimates - fractional Sobolev norms
    • Corner singularities
    • EWI-16210

    Cite this

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    title = "Stability estimates for h-p spectral element methods for general elliptic problems on curvilinear domains",
    abstract = "In this paper we show that the h-p spectral element method developed in 3,8,9 applies to elliptic problems in curvilinear polygons with mixed Neumann and Dirichlet boundary conditions provided that the Babuska–Brezzi inf-sup conditions are satisfied. We establish basic stability estimates for a non-conforming h-p spectral element method which allows for simultaneous mesh refinement and variable polynomial degree. The spectral element functions are non-conforming if the boundary conditions are Dirichlet. For problems with mixed boundary conditions they are continuous only at the vertices of the elements. We obtain a stability estimate when the spectral element functions vanish at the vertices of the elements, which is needed for parallelizing the numerical scheme. Finally, we indicate how the mesh refinement strategy and choice of polynomial degree depends on the regularity of the coefficients of the differential operator, smoothness of the sides of the polygon and the regularity of the data to obtain the maximum accuracy achievable.",
    keywords = "IR-70531, curvilinear polygons, geometrical mesh, METIS-213783, mixed Neumann Dirichlet boundary conditions, fractional Sobolev norms, infsup conditions, stability estimates, Corner singularities - geometrical mesh - mixed Neumann and Dirichlet boundary conditions - curvilinear polygons - inf-sup conditions - stability estimates - fractional Sobolev norms, Corner singularities, EWI-16210",
    author = "Pravir Dutt and Satyendra Tomar",
    note = "The research by PD is partly supported by Aeronautical Research and Development Board (ARDB) and Center for Development of Advanced Computing (CDAC), Pune, India. The research by ST is supported by Council for Scientific and Industrial Research (CSIR), India.",
    year = "2003",
    month = "11",
    doi = "10.1007/BF02829633",
    language = "English",
    volume = "113",
    pages = "395--429",
    journal = "Proceedings of the Indian academy of sciences - mathematical sciences",
    issn = "0253-4142",
    publisher = "Indian Academy of Sciences",
    number = "4",

    }

    Stability estimates for h-p spectral element methods for general elliptic problems on curvilinear domains. / Dutt, Pravir; Tomar, Satyendra.

    In: Proceedings of the Indian academy of sciences - mathematical sciences, Vol. 113, No. 4, 11.2003, p. 395-429.

    Research output: Contribution to journalArticleAcademicpeer-review

    TY - JOUR

    T1 - Stability estimates for h-p spectral element methods for general elliptic problems on curvilinear domains

    AU - Dutt, Pravir

    AU - Tomar, Satyendra

    N1 - The research by PD is partly supported by Aeronautical Research and Development Board (ARDB) and Center for Development of Advanced Computing (CDAC), Pune, India. The research by ST is supported by Council for Scientific and Industrial Research (CSIR), India.

    PY - 2003/11

    Y1 - 2003/11

    N2 - In this paper we show that the h-p spectral element method developed in 3,8,9 applies to elliptic problems in curvilinear polygons with mixed Neumann and Dirichlet boundary conditions provided that the Babuska–Brezzi inf-sup conditions are satisfied. We establish basic stability estimates for a non-conforming h-p spectral element method which allows for simultaneous mesh refinement and variable polynomial degree. The spectral element functions are non-conforming if the boundary conditions are Dirichlet. For problems with mixed boundary conditions they are continuous only at the vertices of the elements. We obtain a stability estimate when the spectral element functions vanish at the vertices of the elements, which is needed for parallelizing the numerical scheme. Finally, we indicate how the mesh refinement strategy and choice of polynomial degree depends on the regularity of the coefficients of the differential operator, smoothness of the sides of the polygon and the regularity of the data to obtain the maximum accuracy achievable.

    AB - In this paper we show that the h-p spectral element method developed in 3,8,9 applies to elliptic problems in curvilinear polygons with mixed Neumann and Dirichlet boundary conditions provided that the Babuska–Brezzi inf-sup conditions are satisfied. We establish basic stability estimates for a non-conforming h-p spectral element method which allows for simultaneous mesh refinement and variable polynomial degree. The spectral element functions are non-conforming if the boundary conditions are Dirichlet. For problems with mixed boundary conditions they are continuous only at the vertices of the elements. We obtain a stability estimate when the spectral element functions vanish at the vertices of the elements, which is needed for parallelizing the numerical scheme. Finally, we indicate how the mesh refinement strategy and choice of polynomial degree depends on the regularity of the coefficients of the differential operator, smoothness of the sides of the polygon and the regularity of the data to obtain the maximum accuracy achievable.

    KW - IR-70531

    KW - curvilinear polygons

    KW - geometrical mesh

    KW - METIS-213783

    KW - mixed Neumann Dirichlet boundary conditions

    KW - fractional Sobolev norms

    KW - infsup conditions

    KW - stability estimates

    KW - Corner singularities - geometrical mesh - mixed Neumann and Dirichlet boundary conditions - curvilinear polygons - inf-sup conditions - stability estimates - fractional Sobolev norms

    KW - Corner singularities

    KW - EWI-16210

    U2 - 10.1007/BF02829633

    DO - 10.1007/BF02829633

    M3 - Article

    VL - 113

    SP - 395

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    JO - Proceedings of the Indian academy of sciences - mathematical sciences

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    SN - 0253-4142

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    ER -