Finite elements approximation of second order linear elliptic equations in divergence form with right-hand side in L¹

In this lecture I will report on joint work with J. Casado-Díaz, T. Chacáon Rebollo, V. Girault and M.~Gómez Marmol which was published in Numerische Mathematik, vol. 105, (2007), pp. 337-510.

We consider, in dimension $d\ge 2$, the standard $P^1$ finite elements approximation of the second order linear elliptic equation in divergence form with coefficients in $L^\infty(\Omega)$ 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 $L^1(\Omega)$, we prove that the unique solution of the discrete problem converges in $W^{1,q}_0(\Omega)$ (for every $q$ with $1 \leq q $ < $ {d \over d-1} $) 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 $W^{1,q}_0(\Omega)$ when the right-hand side belongs to $L^r(\Omega)$ for some $r$ > $1$.