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

Abstract

In this paper we consider, in dimension d≥ 2, the standard P1P1 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 L1(Ω), we prove that the unique solution of the discrete problem converges in W1,q0(Ω)W01,q(Ω) (for every q with 1≤q<d/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 W1,q0(Ω) when the right-hand side belongs to Lr(Ω) for some r > 1.