Optimal convergence estimates for the trace of the polynomial 𝐿²-projection operator on a simplex

Chernov, Alexey

doi:10.1090/S0025-5718-2011-02513-5

Optimal convergence estimates for the trace of the polynomial $L^2$-projection operator on a simplex
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Math. Comp. 81 (2012), 765-787 Request permission

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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