Convergence of nonlinear numerical approximations for an elliptic linear problem with irregular data

International audienceThis work is devoted to the study of the approximation, using two nonlinear numerical methods, of alinear elliptic problem with measure data and heterogeneous anisotropic diffusion matrix. Both methodsshow convergence properties to a continuous solution of the problem in a weak sense, through the changeof variable u = ψ(v), where ψ is a well chosen diffeomorphism between (−1, 1) and R, and v is valuedin (−1, 1). We first study a nonlinear finite element approximation on any simplicial grid. We prove theexistence of a discrete solution, and, under standard regularity conditions, we prove its convergence to aweak solution of the problem by applying Hölder and Sobolev inequalities. Some numerical results, in 2Dand 3D cases where the solution does not belong to H 1 (Ω), show that this method can provide accurateresults. We then construct a numerical scheme which presents a convergence property to the entropyweak solution of the problem in the case where the right-hand side belongs to L 1 . This is achieved owingto a nonlinear control volume finite element (CVFE) method, keeping the same nonlinear reformulation,and adding an upstream weighting evaluation and a nonlinear p−Laplace vanishing stabilisation term