Optimal error estimates in Jacobi-weighted Sobolev spaces for polynomial approximations on the triangle

Li, Huiyuan; Shen, Jie

doi:10.1090/S0025-5718-09-02308-4

Optimal error estimates in Jacobi-weighted Sobolev spaces for polynomial approximations on the triangle
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by Huiyuan Li and Jie Shen;

Math. Comp. 79 (2010), 1621-1646

DOI: https://doi.org/10.1090/S0025-5718-09-02308-4

Published electronically: September 17, 2009

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Abstract:

Spectral approximations on the triangle by orthogonal polynomials are studied in this paper. Optimal error estimates in weighted semi-norms for both the $L^2-$ and $H^1_0-$orthogonal polynomial projections are established by using the generalized Koornwinder polynomials and the properties of the Sturm-Liouville operator on the triangle. These results are then applied to derive error estimates for the spectral-Galerkin method for second- and fourth-order equations on the triangle. The generalized Koornwinder polynomials and approximation results developed in this paper will be useful for many other applications involving spectral and spectral-element approximations in triangular domains.

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Optimal error estimates in Jacobi-weighted Sobolev spaces for polynomial approximations on the triangle
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Abstract: