Numerical solution of a Laplace equation with data in L1

Abstract

In this work, we address the numerical solution of the Laplace equation with data in L1 by IP1 finite element schemes. Even if this is a simple problem, its analysis is difficult and requires new tools because finite element schemes are based on variational formulations which do not lend themselves to estimates in the L1 norm. The approach for analyzing this problem consists in applying some of the techniques that are used by Murat (cf. [5]) and Boccardo & Gallouet (cf. [2]) in constructing the renormalized solution of the problem. The key ingredient is the assumption that all the angles of the grid are acute; then the matrix of the system is an M matrix. Interestingly, with this sole assumption, we prove that uh tends to u in mesure in Ω.