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Optimal convergence estimates for the trace of the polynomial $ L^2$-projection operator on a simplex

Author: Alexey Chernov
Journal: Math. Comp. 81 (2012), 765-787
MSC (2010): Primary 41A10, 65N35, 41A25, 65N15
Published electronically: June 7, 2011
MathSciNet review: 2869036
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Abstract: In this paper we study convergence of the $ L^2$-projection onto the space of polynomials up to degree $ p$ on a simplex in $ \mathbb{R}^d$, $ d \geq 2$. Optimal error estimates are established in the case of Sobolev regularity and illustrated on several numerical examples. The proof is based on the collapsed coordinate transform and the expansion into various polynomial bases involving Jacobi polynomials and their antiderivatives. The results of the present paper generalize corresponding estimates for cubes in $ \mathbb{R}^d$ from [P. Houston, C. Schwab, E. Süli, Discontinuous $ hp$-finite element methods for advection-diffusion-reaction problems. SIAM J. Numer. Anal. 39 (2002), no. 6, 2133-2163].

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Additional Information

Alexey Chernov
Affiliation: Hausdorff Center for Mathematics & Institute for Numerical Simulation, University of Bonn, 53115 Bonn, Germany

Keywords: $L^{2}$-projection, simplex, orthogonal polynomials, error estimate, $p$-version, spectral method
Received by editor(s): April 12, 2010
Received by editor(s) in revised form: December 27, 2010
Published electronically: June 7, 2011
Additional Notes: The author acknowledges support by the Hausdorff Center for Mathematics
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.