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Convergence of Cubic Renormalized FEM for Linear Elliptic Equations with L1-Data |
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PP: 933-945 |
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doi:10.18576/amis/180502
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Author(s) |
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Rachid Messaoudi,
Abdeluaab Lidouh,
Shawkat Alkhazaleh,
Mohamed A. Hafez,
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Abstract |
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| This paper contains a study based on the usual \textbf{cubic} FEM by which we extend all results obtained in \cite{Murat-1,Rachid-1} where the case of \textbf{$P_{n}$} $(n=1, 2)$ renormalized FEM is considered to approximate the solution of linear elliptic equation with $L^{\infty}\left( \Omega\right)$-coefficients, $L^1$-data and which generalizes Laplaces equation.\By introducing a same techniques adopted in \cite{Murat-1,Rachid-1}, where the dimension is $d = 2, 3$, the convergence for the unique discrete solution in $W_{0}^{1,q}\left( \Omega\right)$ for every $q \in [ 1,d/(d-1)[ $ to the unique renormalized solution of the problem are proved, and the estimates of the error are derived. Thereby, similarly as the previous studies, in the case of a bounded Radon measure data, a weaker result is obtained. An error estimate in $W_{0}^{1,q}\left(\Omega\right) $ for smooth coefficients and $L^{r}\left(\Omega\right) $-data such that $T_{k}\left( f\right) \in H^{1}\left(\Omega\right) $ for all $(k,r)$ in $\mathbb R^{+*}\times]1,\infty[,$ is given.
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