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

Abstract

This work is devoted to the study of the approximation, using two nonlinear numerical methods, of a  linear elliptic problem with measure data and heterogeneous anisotropic diffusion matrix. Both methods  show convergence properties to a continuous solution of the problem in a weak sense, through the change  of variable u = ψ(v), where ψ is a well chosen diffeomorphism between (−1, 1) and R, and v is valued  in (−1, 1). We first study a nonlinear finite element approximation on any simplicial grid. We prove the existence of a discrete solution, and, under standard regularity conditions, we prove its convergence to a  weak solution of the problem by applying Hölder and Sobolev inequalities. Some numerical results, in 2D  and 3D cases where the solution does not belong to H 1(Ω), show that this method can provide accurate  results. We then construct a numerical scheme which presents a convergence property to the entropy  weak solution of the problem in the case where the right-hand side belongs to L1 . This is achieved owing  to 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.