Volume 7, Issue 3
Extrapolation of Finite Element Approximation in a Rectangular Domain
DOI:

J. Comp. Math., 7 (1989), pp. 227-233.

Published online: 1989-07

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• Abstract

Recently, the Richardson extrapolation for the elliptic Ritz projection with linear triangular elements on a general convex polygonal domain was discussed by Lin and Lu. We go back in this note to the simplest case, i.e. the bilinear rectangular elements on a rectangular domain which is a parallel case of the one-triangle model in the early work of Lin and Liu. We find that the finite element argument for the Richardson extrapolation with an accuracy of $O(h^4)$ needs only the regularity of $H^{4,\infty}$ for the solution u but the finite difference argument for extrapolation with $O(h^{s+\alpha})$ accuracy needs $u\in C^{5+\alpha}(0‹\alpha‹1)$. Moreover, a formula is suggested to guarantee the extrapolation of $O(h^4)$ accuracy at fine gridpoints as well as at coarse gridpoints.

• Keywords

@Article{JCM-7-227, author = {}, title = {Extrapolation of Finite Element Approximation in a Rectangular Domain}, journal = {Journal of Computational Mathematics}, year = {1989}, volume = {7}, number = {3}, pages = {227--233}, abstract = { Recently, the Richardson extrapolation for the elliptic Ritz projection with linear triangular elements on a general convex polygonal domain was discussed by Lin and Lu. We go back in this note to the simplest case, i.e. the bilinear rectangular elements on a rectangular domain which is a parallel case of the one-triangle model in the early work of Lin and Liu. We find that the finite element argument for the Richardson extrapolation with an accuracy of $O(h^4)$ needs only the regularity of $H^{4,\infty}$ for the solution u but the finite difference argument for extrapolation with $O(h^{s+\alpha})$ accuracy needs $u\in C^{5+\alpha}(0‹\alpha‹1)$. Moreover, a formula is suggested to guarantee the extrapolation of $O(h^4)$ accuracy at fine gridpoints as well as at coarse gridpoints. }, issn = {1991-7139}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/jcm/9473.html} }
TY - JOUR T1 - Extrapolation of Finite Element Approximation in a Rectangular Domain JO - Journal of Computational Mathematics VL - 3 SP - 227 EP - 233 PY - 1989 DA - 1989/07 SN - 7 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/jcm/9473.html KW - AB - Recently, the Richardson extrapolation for the elliptic Ritz projection with linear triangular elements on a general convex polygonal domain was discussed by Lin and Lu. We go back in this note to the simplest case, i.e. the bilinear rectangular elements on a rectangular domain which is a parallel case of the one-triangle model in the early work of Lin and Liu. We find that the finite element argument for the Richardson extrapolation with an accuracy of $O(h^4)$ needs only the regularity of $H^{4,\infty}$ for the solution u but the finite difference argument for extrapolation with $O(h^{s+\alpha})$ accuracy needs $u\in C^{5+\alpha}(0‹\alpha‹1)$. Moreover, a formula is suggested to guarantee the extrapolation of $O(h^4)$ accuracy at fine gridpoints as well as at coarse gridpoints.