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

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, $W^{1,\infite}$-interior estimate $\|u_n\|_{1,\infite,\Omega_0$\leq c\|u_h\|_{-s,\Omega_1},u_h\in S^h(\Omega_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,\infite}$-interior error estimate$\|u-u_n\|_{1,\infite,\Omega_0$\leq c h^{r-1} (\|u\|_{r,\infite,\Omega_1}+\|u\|_{1,\Omega}$ If the triangulation is strongly regular in $\Omega_1$ and r=2 we obtain $W^{1,\infite}$-interior superconvergence.

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@Article{JCM-3-1, author = {H. T. Banks, S. H. Hu and Z. R. Kenz}, title = {W^{1,infty}-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, $W^{1,\infite}$-interior estimate $\|u_n\|_{1,\infite,\Omega_0$\leq c\|u_h\|_{-s,\Omega_1},u_h\in S^h(\Omega_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,\infite}$-interior error estimate$\|u-u_n\|_{1,\infite,\Omega_0$\leq c h^{r-1} (\|u\|_{r,\infite,\Omega_1}+\|u\|_{1,\Omega}$ If the triangulation is strongly regular in $\Omega_1$ and r=2 we obtain $W^{1,\infite}$-interior superconvergence. }, issn = {1991-7139}, doi = {https://doi.org/10.4208/aamm.10-m1030}, url = {http://global-sci.org/intro/article_detail/jcm/9601.html} }
TY - JOUR T1 - W^{1,infty}-Interior Estimates for Finite Element Method on Regular Mesh AU - H. T. Banks, S. H. Hu & Z. R. Kenz JO - Journal of Computational Mathematics VL - 1 SP - 1 EP - 7 PY - 1985 DA - 1985/03 SN - 3 DO - http://doi.org/10.4208/aamm.10-m1030 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, $W^{1,\infite}$-interior estimate $\|u_n\|_{1,\infite,\Omega_0$\leq c\|u_h\|_{-s,\Omega_1},u_h\in S^h(\Omega_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,\infite}$-interior error estimate$\|u-u_n\|_{1,\infite,\Omega_0$\leq c h^{r-1} (\|u\|_{r,\infite,\Omega_1}+\|u\|_{1,\Omega}$ If the triangulation is strongly regular in $\Omega_1$ and r=2 we obtain $W^{1,\infite}$-interior superconvergence.