Finite elements approximation of second order linear elliptic equations in divergence form with right-hand side in L 1

Abstract

In this paper we consider, in dimension d≥ 2, the standard $$\mathbb{P}_{1}$$ 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 $$W^{1,q}_0(\Omega)$$ (for every q with $${1 \leq q 1.