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h-p Spectral Element Method for Elliptic Problems on Non-smooth Domains Using Parallel Computers

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Abstract

We propose a new h-p spectral element method to solve elliptic boundary value problems with mixed Neumann and Dirichlet boundary conditions on non-smooth domains. The method is shown to be exponentially accurate and asymptotically faster than the standard h-p finite element method. The spectral element functions are fully non-conforming for pure Dirichlet problems and conforming only at the vertices of the elements for mixed problems, and hence, the dimension of the resulting Schur complement matrix is quite small. The method is a least-squares collocation method and the resulting normal equations are solved using preconditioned conjugate gradient method with an almost optimal preconditioner. The algorithm is suitable for a distributed memory parallel computer. The numerical results of a number of model problems are presented, which confirm the theoretical estimates.

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References

  • I. Babuska A. Craig J. Mandel J. Pitkäranta (1991) ArticleTitleEfficient preconditioning for the p version of the finite element method in two dimensions SIAM J. Num. Anal 28 IssueID3 624 Occurrence Handle0754.65083 Occurrence Handle10.1137/0728034

    Article  MATH  Google Scholar 

  • I. Babuska B. Q. Guo (1988) ArticleTitleRegularity of the solution of elliptic problems with piecewise analytic data. part-I SIAM J. Math. Anal 19 172 Occurrence Handle0647.35021 Occurrence Handle924554 Occurrence Handle10.1137/0519014

    Article  MATH  MathSciNet  Google Scholar 

  • I. Babuska B. Q. Guo (1988) ArticleTitleThe h-p version of the finite element method on domains with curved boundaries SIAM J. Num. Anal. 25 837 Occurrence Handle0655.65124 Occurrence Handle954788 Occurrence Handle10.1137/0725048

    Article  MATH  MathSciNet  Google Scholar 

  • I. Babuska H.-S. Oh (1990) ArticleTitleThe p-version of the finite element method for domains with corners and for infinite domains Num. Meth. PDEs 6 371 Occurrence Handle0717.65084 Occurrence Handle1087251

    MATH  MathSciNet  Google Scholar 

  • I. Babuska M. Suri (1994) ArticleTitleThe p and h-p versions of the finite element method, basic principles and properties SIAM Rev. 36 IssueID4 578 Occurrence Handle0813.65118 Occurrence Handle1306924 Occurrence Handle10.1137/1036141

    Article  MATH  MathSciNet  Google Scholar 

  • P. Dutt (1990) ArticleTitleSpectral methods for initial boundary value problems – an alternative approach SIAM J. Num. Anal. 27 IssueID4 885 Occurrence Handle0708.65089 Occurrence Handle1051112 Occurrence Handle10.1137/0727051

    Article  MATH  MathSciNet  Google Scholar 

  • P. K. Dutt A. K. Singh (1994) ArticleTitleThe Galerkin-collocation method for hyperbolic initial boundary value problems J. Comp. Phys. 112 IssueID2 211 Occurrence Handle0809.65100 Occurrence Handle1277274 Occurrence Handle10.1006/jcph.1994.1093

    Article  MATH  MathSciNet  Google Scholar 

  • P. Dutt S. Joshi (2001) ArticleTitleSpectral methods for hyperbolic initial boundary value problems on parallel computers J. Comp. Appl. Math. 134 IssueID1–2 165 Occurrence Handle0987.65101 Occurrence Handle10.1016/S0377-0427(00)00535-5

    Article  MATH  Google Scholar 

  • P. Dutt S. Tomar R. Kumar (2002) ArticleTitleStability estimates for h-p spectral element methods for elliptic problems Proc. Indian Acad. Sci. (Math. Sci.) 112(4) 601 Occurrence Handle10.1007/BF02829693

    Article  Google Scholar 

  • P. Dutt S. Tomar (2003) ArticleTitleStability estimates for h-p spectral element methods for general elliptic problems on curvilinear domains Proc. Indian Acad. Sci. (Math. Sci.) 113 IssueID4 395 Occurrence Handle1045.65095 Occurrence Handle2020074

    MATH  MathSciNet  Google Scholar 

  • B. Guo W. Cao (1996) ArticleTitleA preconditioner for the h-p version of the finite element method in two dimensions Numer. Math. 75 59 Occurrence Handle0873.65047 Occurrence Handle1417863 Occurrence Handle10.1007/s002110050230

    Article  MATH  MathSciNet  Google Scholar 

  • Karniadakis, G., Spencer, S. J.: Spectral/hp-element methods for CFD. Oxford University Press 1999.

  • Kondratiev, V. A.: The smoothness of a solution of Dirichlet's problem for second-order elliptic equations in a region with a piecewise smooth boundary. Differential' nye Uraneniya 6(10), 1831 (and Differential Equations 6, 1392, 1970).

    Google Scholar 

  • D. Pathria G. E. Karniadakis (1995) ArticleTitleSpectral element methods for elliptic problems in nonsmooth domains J. Comp. Phys. 122 83 Occurrence Handle0844.65082 Occurrence Handle1358523 Occurrence Handle10.1006/jcph.1995.1198

    Article  MATH  MathSciNet  Google Scholar 

  • Schoeberl, J., Melenk, J. M., Pechstein, C., Zaglmayr, S.: Additive schwarz preconditioning for p-version triangular and tetrahedral finite elements. Tech. Report 2005-11, RICAM, 2005.

  • Ch. Schwab (1998) p- and hp-finite element methods Clarendon Press Oxford

    Google Scholar 

  • Tomar, S. K.: h-p spectral element methods for elliptic problems on non-smooth domains using parallel computers, Ph.D. thesis, IIT Kanpur, India (2001). Reprint available as Tech. Rep. no. 1631, Faculty of Mathematical Sciences, University of Twente, The Netherlands.

  • Tomar, S. K., Dutt, P., Rathish Kumar, B. V.: An efficient and exponentially accurate parallel h-p spectral element method for elliptic problems on polygonal domains – the Dirichlet case. In: Proc. 9th Int. Conf. High Performance Computing. Lecture Notes in Computer Science, Vol. 2552, pp. 534. Springer 2002.

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Tomar, S.K. h-p Spectral Element Method for Elliptic Problems on Non-smooth Domains Using Parallel Computers. Computing 78, 117–143 (2006). https://doi.org/10.1007/s00607-006-0176-0

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