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

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M3 - Article

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

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