In this lecture I will report on joint work with J. Casado-Diaz, T.
Chacon Rebollo, V. Girault and M. Gomez Marmol.
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
$/displaystyle{1 /leq q<{d /over d-1}})$ 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 $r>1$.