Local superconvergence of the derivative for tensor-product block FEM

• Published in: Numerical methods for partial differential equations Vol. 28; no. 2; pp. 457 - 475
• Main Authors: He, Wen-Ming, Chen, Wei-Qiu, Zhu, Qi-Ding
• Format: Journal Article
• Language: English
• Published: Hoboken Wiley Subscription Services, Inc., A Wiley Company 01.03.2012
• Subjects: finite element theory of Green function; local superconvergence; tensor-product block finite elements; the symmetry technique; weak estimate
• ISSN: 0749-159X; 1098-2426
• DOI: 10.1002/num.20628

In this article, we shall discuss local superconvergence of the derivative for tensor‐product block finite elements over uniform partition for three‐dimensional Poisson's equation on the basis of Liu and Zhu (Numer Methods Partial Differential Eq 25 (2009) 999–1008). Assume that odd m ≥ 3, x0 is an inner locally symmetric point of uniform rectangular partition \documentclass{article} \usepackage{mathrsfs} \usepackage{amsmath, amssymb} \pagestyle{empty} \begin{document} \begin{align*}\mathcal{T}_{h}\end{align*} \end{document} and ρ(x0,∂Ω) means the distance between x0 and boundary ∂Ω. Combining the symmetry technique (Wahlbin, Springer, 1995; Schatz, Sloan, and Wahlbin, SIAM J Numer Anal 33 (1996), 505–521; Schatz, Math Comput 67 (1998), 877–899) with weak estimates for tensor‐product block finite elements of degree m ≥ 3 [see Liu and Zhu, Numer Methods Partial Differential Eq 25 (2009) 999–1008] and the finite element theory of Green function in ℜ︁3 presented in this article, we propose the \documentclass{article} \usepackage{mathrsfs} \usepackage{amsmath, amssymb} \pagestyle{empty} \begin{document} \begin{align*}O(h^{m+3}|\ln h|^{\frac{4}{3}}+h^{2m+2}|\ln h|^{\frac{4}{3}}\rho(x_{0},\partial\Omega)^{-m})\end{align*} \end{document} convergence of the derivatives for tensor‐product block finite elements of degree m ≥ 3 on x0. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 28: 457–475, 2012