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Error analysis of spectral method on a triangle

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Abstract

In this paper, the orthogonal polynomial approximation on triangle, proposed by Dubiner, is studied. Some approximation results are established in certain non-uniformly Jacobi-weighted Sobolev space, which play important role in numerical analysis of spectral and triangle spectral element methods for differential equations on complex geometries. As an example, a model problem is considered.

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References

  1. P. Appell and J. Kampé de Fériet, Functions hypergéométriques et hyperspheriques-polynomes d'Hermite (Gauthier-Villars, Paris, 1926).

    Google Scholar 

  2. I. Babuška and B.Q. Guo, Optimal estimates for lower and upper bounds of approximation errors in the p-version of the finite element method in two-dimensions, Numer. Math. 85 (2000) 219–255.

    Article  MathSciNet  MATH  Google Scholar 

  3. I. Babuška and B.Q. Guo, Direct and inverse approximation theorems for the p-version of the finite element method in the framework of weighted Besov spacecs, part I, approximability of functions in weighted Besov spaces, SIAM J. Numer. Anal. 39 (2001) 1512–1538.

    Article  MathSciNet  MATH  Google Scholar 

  4. C. Bernardi and Y. Maday, Spectral method, in: Handbook of Numerical Analysis, Part 5, eds. P.G. Ciarlet and J.L. Lions (North-Holland, Amsterdam, 1977).

    Google Scholar 

  5. W. Cai, High-order mixed current bais functions for electromagnetic scattering of curve surface, J. Sci. Comput. 14 (1999) 73–105.

    Article  MATH  MathSciNet  Google Scholar 

  6. C. Canuto and A. Quarteroni, Approximation results for orthogonal polynomials in Sobolev spaces, Math. Comp. 38 (1982) 67–86.

    Article  MATH  MathSciNet  Google Scholar 

  7. M. Dubiner, Spectral methods on triangles and other domains, J. Sci. Comput. 6 (1991) 345–390.

    Article  MATH  MathSciNet  Google Scholar 

  8. C.F. Dunkl and Y. Xu, Orthogonal Polynomials of Several Variables (Cambridge Univ. Press, Cambridge, 2001).

    MATH  Google Scholar 

  9. D. Funaro, Pseudospectral approximation of a PDE defined on a triangle, Appl. Math. Comput. 42 (1991) 121–138.

    Article  MATH  MathSciNet  Google Scholar 

  10. D. Funaro, Polynomial Approximations of Differential Equations (Springer-Verlag, Berlin, 1992).

    Google Scholar 

  11. B.-y. Guo, Jacobi approximations in certain Hilbert spaces and their applications to singular differential equations, J. Math. Anal. Appl. 243 (2000) 373–408.

    Article  MATH  MathSciNet  Google Scholar 

  12. B.-y. Guo and L.-L. Wang, Non-isotropic Jacobi spectral method, in: Contemporary Mathematics, Vol. 329 (Amer. Math. Soc., Providence, RI, 2003) pp. 157–169.

    Google Scholar 

  13. G.H. Hardy, J.E. Littlewood and G. Pólya, Inequalities (Cambridge Univ. Press, Cambridge, 1952).

    MATH  Google Scholar 

  14. W. Heinrichs, Spectral collocation on triangular elements, J. Comput. Phys. 145 (1998) 743–757.

    Article  MATH  MathSciNet  Google Scholar 

  15. J.S. Hesthaven and D. Gottlieb, Stable spectral methods for conservation law on triangles with unstructured grids, Comput. Methods Appl. Mech. Engrg. 175 (1999) 361–381.

    Article  MATH  MathSciNet  Google Scholar 

  16. G.E. Karniadakis and S.J. Sherwin, Spectral/hp Element Methods for CFD (Oxford Univ. Press, Oxford, 1999).

    MATH  Google Scholar 

  17. R.G. Owens, Spectral Approximation on the triangle, Proc. Roy. Soc. London A 454 (1998) 857–872.

    Article  MATH  MathSciNet  Google Scholar 

  18. S.J. Sherwin and G.E. Farniadakis, A new triangular and tetrahedral basis for high-order finite element methods, Internat. J. Numer. Methods Engrg. 38 (1995) 3775–3802.

    Article  MATH  MathSciNet  Google Scholar 

  19. S.J. Sherwin and G.E. Farniadakis, A triangular spetral element method: applications to the incompressible Navier-Stokes equations, Comput. Methods Appl. Mech. Engrg. 123 (1995) 189–229.

    Article  MATH  MathSciNet  Google Scholar 

  20. G. Szegö, Othogonal Polynomials, Vol. 23, 4th edn. (AWS Coll. Publ., 1975).

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Authors and Affiliations

  1. Department of Mathematics, Shanghai Normal University, Shanghai, 200234, China

    Ben-yu Guo

  2. Division of Computational Science of E-institute of Shanghai Universities, Shanghai, China

    Ben-yu Guo

  3. Department of Mathematics, Nan yang Technology University, 637616, Singapore

    Li-Lian Wang

Authors
  1. Ben-yu Guo
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  2. Li-Lian Wang
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Corresponding author

Correspondence to Ben-yu Guo.

Additional information

Communicated by Z.Y. Chen

Mathematics subject classifications (2000)

33C45, 41A10, 41A25, 65N35

Ben-yu Guo: The work of this author is supported in part by NSF of China, N.10471095, Science Foundation of Shanghai, N. 04JC14062, The Special Funds for Doctorial Authorities of Education Ministry of China, N. 20040270002, E-institutes of Shanghai Municipal Education Commission, N.E03004, The Shanghai Leading Academic Discipline Project N. T0401 and The Fund N.04DB15 of Shanghai Education Commission.

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Guo, By., Wang, LL. Error analysis of spectral method on a triangle. Adv Comput Math 26, 473–496 (2007). https://doi.org/10.1007/s10444-005-7471-8

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  • DOI: https://doi.org/10.1007/s10444-005-7471-8

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