An Efficient and Exponentially Accurate Parallel h-p Spectral Element Method for Elliptic Problems on Polygonal Domains - The Dirichlet Case

S.K. Tomar, P. Dutt, B.V. Ratish Kumar

Research output: Chapter in Book/Report/Conference proceedingConference contributionAcademicpeer-review

6 Citations (Scopus)
44 Downloads (Pure)

Abstract

For smooth problems spectral element methods (SEM) exhibit exponential convergence and have been very successfully used in practical problems. However, in many engineering and scientific applications we frequently encounter the numerical solutions of elliptic boundary value problems in non-smooth domains which give rise to singularities in the solution. In such cases the accuracy of the solution obtained by SEM deteriorates and they offer no advantages over low order methods. A new Parallel h-p Spectral Element Method is presented which resolves this form of singularity by employing a geometric mesh in the neighborhood of the corners and gives exponential convergence with asymptotically faster results than conventional methods. The normal equations are solved by the Preconditioned Conjugate Gradient (PCG) method. Except for the assemblage of the resulting solution vector, all computations are done on the element level and we don't need to compute and store mass and stiffness like matrices. The technique to compute the preconditioner is quite simple and very easy to implement. The method is based on a parallel computer with distributed memory and the library used for message passing is MPI. Load balancing issues are discussed and the communication involved among the processors is shown to be quite small.
Original languageEnglish
Title of host publicationHigh Performance Computing — HiPC 2002
Subtitle of host publication9th International Conference Bangalore, India, December 18–21, 2002 Proceedings
EditorsSartaj Sahni, Viktor K. Prasanna, Uday Shukla
Place of PublicationBerlin, Germany
PublisherSpringer
Pages534-544
Number of pages11
ISBN (Electronic)978-3-540-36265-4
ISBN (Print)978-3-540-00303-8
DOIs
Publication statusPublished - 2002

Publication series

NameLecture Notes In Computer Science
PublisherSpringer
Volume2552
ISSN (Print)0302-9743

Fingerprint

Spectral Element Method
Elliptic Problems
Dirichlet
Exponential Convergence
Singularity
Nonsmooth Domains
Normal Equations
Preconditioned Conjugate Gradient Method
Elliptic Boundary Value Problems
Distributed Memory
Parallel Computers
Message Passing
Load Balancing
Preconditioner
Stiffness
Resolve
Numerical Solution
Mesh
Engineering

Keywords

  • METIS-207967
  • EWI-16263
  • IR-74817

Cite this

Tomar, S. K., Dutt, P., & Ratish Kumar, B. V. (2002). An Efficient and Exponentially Accurate Parallel h-p Spectral Element Method for Elliptic Problems on Polygonal Domains - The Dirichlet Case. In S. Sahni, V. K. Prasanna, & U. Shukla (Eds.), High Performance Computing — HiPC 2002: 9th International Conference Bangalore, India, December 18–21, 2002 Proceedings (pp. 534-544). (Lecture Notes In Computer Science; Vol. 2552). Berlin, Germany: Springer. https://doi.org/10.1007/3-540-36265-7_50
Tomar, S.K. ; Dutt, P. ; Ratish Kumar, B.V. / An Efficient and Exponentially Accurate Parallel h-p Spectral Element Method for Elliptic Problems on Polygonal Domains - The Dirichlet Case. High Performance Computing — HiPC 2002: 9th International Conference Bangalore, India, December 18–21, 2002 Proceedings. editor / Sartaj Sahni ; Viktor K. Prasanna ; Uday Shukla. Berlin, Germany : Springer, 2002. pp. 534-544 (Lecture Notes In Computer Science).
@inproceedings{dbeeae4dae614008857ce14de635bfbd,
title = "An Efficient and Exponentially Accurate Parallel h-p Spectral Element Method for Elliptic Problems on Polygonal Domains - The Dirichlet Case",
abstract = "For smooth problems spectral element methods (SEM) exhibit exponential convergence and have been very successfully used in practical problems. However, in many engineering and scientific applications we frequently encounter the numerical solutions of elliptic boundary value problems in non-smooth domains which give rise to singularities in the solution. In such cases the accuracy of the solution obtained by SEM deteriorates and they offer no advantages over low order methods. A new Parallel h-p Spectral Element Method is presented which resolves this form of singularity by employing a geometric mesh in the neighborhood of the corners and gives exponential convergence with asymptotically faster results than conventional methods. The normal equations are solved by the Preconditioned Conjugate Gradient (PCG) method. Except for the assemblage of the resulting solution vector, all computations are done on the element level and we don't need to compute and store mass and stiffness like matrices. The technique to compute the preconditioner is quite simple and very easy to implement. The method is based on a parallel computer with distributed memory and the library used for message passing is MPI. Load balancing issues are discussed and the communication involved among the processors is shown to be quite small.",
keywords = "METIS-207967, EWI-16263, IR-74817",
author = "S.K. Tomar and P. Dutt and {Ratish Kumar}, B.V.",
year = "2002",
doi = "10.1007/3-540-36265-7_50",
language = "English",
isbn = "978-3-540-00303-8",
series = "Lecture Notes In Computer Science",
publisher = "Springer",
pages = "534--544",
editor = "Sartaj Sahni and Prasanna, {Viktor K.} and Uday Shukla",
booktitle = "High Performance Computing — HiPC 2002",

}

Tomar, SK, Dutt, P & Ratish Kumar, BV 2002, An Efficient and Exponentially Accurate Parallel h-p Spectral Element Method for Elliptic Problems on Polygonal Domains - The Dirichlet Case. in S Sahni, VK Prasanna & U Shukla (eds), High Performance Computing — HiPC 2002: 9th International Conference Bangalore, India, December 18–21, 2002 Proceedings. Lecture Notes In Computer Science, vol. 2552, Springer, Berlin, Germany, pp. 534-544. https://doi.org/10.1007/3-540-36265-7_50

An Efficient and Exponentially Accurate Parallel h-p Spectral Element Method for Elliptic Problems on Polygonal Domains - The Dirichlet Case. / Tomar, S.K.; Dutt, P.; Ratish Kumar, B.V.

High Performance Computing — HiPC 2002: 9th International Conference Bangalore, India, December 18–21, 2002 Proceedings. ed. / Sartaj Sahni; Viktor K. Prasanna; Uday Shukla. Berlin, Germany : Springer, 2002. p. 534-544 (Lecture Notes In Computer Science; Vol. 2552).

Research output: Chapter in Book/Report/Conference proceedingConference contributionAcademicpeer-review

TY - GEN

T1 - An Efficient and Exponentially Accurate Parallel h-p Spectral Element Method for Elliptic Problems on Polygonal Domains - The Dirichlet Case

AU - Tomar, S.K.

AU - Dutt, P.

AU - Ratish Kumar, B.V.

PY - 2002

Y1 - 2002

N2 - For smooth problems spectral element methods (SEM) exhibit exponential convergence and have been very successfully used in practical problems. However, in many engineering and scientific applications we frequently encounter the numerical solutions of elliptic boundary value problems in non-smooth domains which give rise to singularities in the solution. In such cases the accuracy of the solution obtained by SEM deteriorates and they offer no advantages over low order methods. A new Parallel h-p Spectral Element Method is presented which resolves this form of singularity by employing a geometric mesh in the neighborhood of the corners and gives exponential convergence with asymptotically faster results than conventional methods. The normal equations are solved by the Preconditioned Conjugate Gradient (PCG) method. Except for the assemblage of the resulting solution vector, all computations are done on the element level and we don't need to compute and store mass and stiffness like matrices. The technique to compute the preconditioner is quite simple and very easy to implement. The method is based on a parallel computer with distributed memory and the library used for message passing is MPI. Load balancing issues are discussed and the communication involved among the processors is shown to be quite small.

AB - For smooth problems spectral element methods (SEM) exhibit exponential convergence and have been very successfully used in practical problems. However, in many engineering and scientific applications we frequently encounter the numerical solutions of elliptic boundary value problems in non-smooth domains which give rise to singularities in the solution. In such cases the accuracy of the solution obtained by SEM deteriorates and they offer no advantages over low order methods. A new Parallel h-p Spectral Element Method is presented which resolves this form of singularity by employing a geometric mesh in the neighborhood of the corners and gives exponential convergence with asymptotically faster results than conventional methods. The normal equations are solved by the Preconditioned Conjugate Gradient (PCG) method. Except for the assemblage of the resulting solution vector, all computations are done on the element level and we don't need to compute and store mass and stiffness like matrices. The technique to compute the preconditioner is quite simple and very easy to implement. The method is based on a parallel computer with distributed memory and the library used for message passing is MPI. Load balancing issues are discussed and the communication involved among the processors is shown to be quite small.

KW - METIS-207967

KW - EWI-16263

KW - IR-74817

U2 - 10.1007/3-540-36265-7_50

DO - 10.1007/3-540-36265-7_50

M3 - Conference contribution

SN - 978-3-540-00303-8

T3 - Lecture Notes In Computer Science

SP - 534

EP - 544

BT - High Performance Computing — HiPC 2002

A2 - Sahni, Sartaj

A2 - Prasanna, Viktor K.

A2 - Shukla, Uday

PB - Springer

CY - Berlin, Germany

ER -

Tomar SK, Dutt P, Ratish Kumar BV. An Efficient and Exponentially Accurate Parallel h-p Spectral Element Method for Elliptic Problems on Polygonal Domains - The Dirichlet Case. In Sahni S, Prasanna VK, Shukla U, editors, High Performance Computing — HiPC 2002: 9th International Conference Bangalore, India, December 18–21, 2002 Proceedings. Berlin, Germany: Springer. 2002. p. 534-544. (Lecture Notes In Computer Science). https://doi.org/10.1007/3-540-36265-7_50