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

David Maltese,Robert Eymard

doi:10.1051/m2an/2021079

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 H1 (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.

Highlights

This paper is devoted to the numerical approximation of a second order linear elliptic equation in divergence form with coefficients in L∞(Ω) and measure data
The convergence of the approximate solution to a weak solution can be proved in the case where the finite element scheme is similar to that used in the finite volume framework, that is when it relies on a Two-Point Flux interpretation of the finite element scheme
The two nonlinear methods presented in this paper are proved to converge to a weak solution of the linear elliptic problem with general measure data and heterogeneous anisotropic diffusion fields, which does not seem to have been proved for earlier schemes

Summary

Introduction

This paper is devoted to the numerical approximation of a second order linear elliptic equation in divergence form with coefficients in L∞(Ω) and measure data. The convergence of the approximate solution to a weak solution (which is the renormalised solution or equivalently the entropy solution in the sense of Def. 3.1) can be proved in the case where the finite element scheme is similar to that used in the finite volume framework, that is when it relies on a Two-Point Flux interpretation of the finite element scheme This requires the use of P 1 finite elements on triangles or tetrahedra which satisfy strong geometrical constraints (in 2D, for two triangles sharing the same edge, the sum of the two opposite angles must be lower than π in the case Λ = Id). The second one concerns a nonlinear Control-Volume Finite-Element scheme (CVFE for short) for which, in addition, a proof of convergence to the entropy weak solution holds in the case where f ∈ L1(Ω)

Motivation and organisation

The numerical scheme

Main results of Section 2

Estimates

Convergence of the nonlinear finite element scheme

Main results of Section 3

Numerical tests

Some open and closed problems

Computation of some integrals in simplices

Equivalence of norms