Finite elements approximation of second order linear elliptic equations in divergence form with right-hand side in L
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Authors: | J Casado-Díaz T Chacón Rebollo V Girault M Gómez Mármol F Murat |
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Institution: | 1. Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Apdo.?de correos 1160, 41080, Sevilla, Spain 2. Laboratoire Jacques-Louis Lions, CNRS UMR 7598, Université Paris VI, Bo?te courrier 187, 75252, Paris cedex 05, France
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Abstract: | In this paper we consider, in dimension d≥ 2, the standard finite elements approximation of the second order linear elliptic equation in divergence form with coefficients in L
∞(Ω) which generalizes Laplace’s equation. We assume that the family of triangulations is regular and that it satisfies an
hypothesis close to the classical hypothesis which implies the discrete maximum principle. When the right-hand side belongs
to L
1(Ω), we prove that the unique solution of the discrete problem converges in (for every q with ) to the unique renormalized solution of the problem. We obtain a weaker result when the right-hand side is a bounded Radon
measure. In the case where the dimension is d = 2 or d = 3 and where the coefficients are smooth, we give an error estimate in when the right-hand side belongs to L
r
(Ω) for some r > 1. |
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Keywords: | Mathematics Subject Classification" target="_blank">Mathematics Subject Classification 65N12 65N30 35A35 35J25 |
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