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

Authors: Huiyuan Li and Jie Shen
Journal: Math. Comp. 79 (2010), 1621-1646
MSC (2000): Primary 65N35, 65N22, 65F05, 35J05
Published electronically: September 17, 2009
MathSciNet review: 2630005
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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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Additional Information

Huiyuan Li
Affiliation: Institute of Software, Chinese Academy of Sciences, Beijing 100190, People’s Republic of China
MR Author ID: 708582

Jie Shen
Affiliation: Department of Mathematics, Purdue University, West Lafayette, Indiana, 47907
MR Author ID: 257933
ORCID: 0000-0002-4885-5732

Keywords: Orthogonal polynomials, Koornwinder polynomials, error estimate, spectral method
Received by editor(s): August 12, 2008
Received by editor(s) in revised form: June 1, 2009
Published electronically: September 17, 2009
Additional Notes: The first author was partially supported by the NSFC grants 10601056, 10431050 and 60573023.
The second author was partially supported by the NFS grant DMS-0610646 and AFOSR FA9550-08-1-0416.
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.