Volume 55, Number 6, November-December 2021
|Page(s)||3043 - 3089|
|Published online||17 December 2021|
Convergence of nonlinear numerical approximations for an elliptic linear problem with irregular data
LAMA, Univ. Gustave Eiffel, Univ. Paris Est Créteil, CNRS, F-77454 Marne-la-Vallée, France
* Corresponding author: firstname.lastname@example.org
Accepted: 25 November 2021
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.
Mathematics Subject Classification: 65N12 / 65N30
Key words: Elliptic equations with irregular data / finite elements / control-volume finite elements / entropy solution
© The authors. Published by EDP Sciences, SMAI 2021
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