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Volume 3, Issue 1
$W^{1,∞}$-Interior Estimates for Finite Element Method on Regular Mesh

Chuan-Miao Chen

J. Comp. Math., 3 (1985), pp. 1-7.

Published online: 1985-03

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  • Abstract

For a large class of piecewise polynomial subspaces $S^h$ defined on the regular mesh, $W^{1,∞}$-interior estimate $\|u_h\|_{1,∞,Ω_0}$ ≤ $c\|u_h\|_{-s,Ω_1}$, $u_h\in S^h{Ω_1}$ satisfying the interior Ritz equation is proved. For the finite element approximation $u_h$ (of degree $r-1$) to $u$, we have $W^{1,∞}$-interior error estimate $\|u-u_h\|_{1,∞,Ω_0}$)≤$ch^{r-1} (\|u\|_{r,∞,Ω_1}+\|u\|_{1,Ω}$). If the triangulation is strongly regular in $Ω_1$ and $r=2$ we obtain $W^{1,∞}$-interior superconvergence.

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@Article{JCM-3-1, author = {Chen , Chuan-Miao}, title = {$W^{1,∞}$-Interior Estimates for Finite Element Method on Regular Mesh}, journal = {Journal of Computational Mathematics}, year = {1985}, volume = {3}, number = {1}, pages = {1--7}, abstract = {

For a large class of piecewise polynomial subspaces $S^h$ defined on the regular mesh, $W^{1,∞}$-interior estimate $\|u_h\|_{1,∞,Ω_0}$ ≤ $c\|u_h\|_{-s,Ω_1}$, $u_h\in S^h{Ω_1}$ satisfying the interior Ritz equation is proved. For the finite element approximation $u_h$ (of degree $r-1$) to $u$, we have $W^{1,∞}$-interior error estimate $\|u-u_h\|_{1,∞,Ω_0}$)≤$ch^{r-1} (\|u\|_{r,∞,Ω_1}+\|u\|_{1,Ω}$). If the triangulation is strongly regular in $Ω_1$ and $r=2$ we obtain $W^{1,∞}$-interior superconvergence.

}, issn = {1991-7139}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/jcm/9601.html} }
TY - JOUR T1 - $W^{1,∞}$-Interior Estimates for Finite Element Method on Regular Mesh AU - Chen , Chuan-Miao JO - Journal of Computational Mathematics VL - 1 SP - 1 EP - 7 PY - 1985 DA - 1985/03 SN - 3 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/jcm/9601.html KW - AB -

For a large class of piecewise polynomial subspaces $S^h$ defined on the regular mesh, $W^{1,∞}$-interior estimate $\|u_h\|_{1,∞,Ω_0}$ ≤ $c\|u_h\|_{-s,Ω_1}$, $u_h\in S^h{Ω_1}$ satisfying the interior Ritz equation is proved. For the finite element approximation $u_h$ (of degree $r-1$) to $u$, we have $W^{1,∞}$-interior error estimate $\|u-u_h\|_{1,∞,Ω_0}$)≤$ch^{r-1} (\|u\|_{r,∞,Ω_1}+\|u\|_{1,Ω}$). If the triangulation is strongly regular in $Ω_1$ and $r=2$ we obtain $W^{1,∞}$-interior superconvergence.

Chen , Chuan-Miao. (1985). $W^{1,∞}$-Interior Estimates for Finite Element Method on Regular Mesh. Journal of Computational Mathematics. 3 (1). 1-7. doi:
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