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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 L 1 . 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.
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https://hal.archives-ouvertes.fr/hal-03105385
Contributor : Robert Eymard Connect in order to contact the contributor
Submitted on : Thursday, November 25, 2021 - 7:21:25 PM
Last modification on : Friday, January 14, 2022 - 3:42:12 AM

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  • HAL Id : hal-03105385, version 2

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Robert Eymard, David Maltese. Convergence of nonlinear numerical approximations for an elliptic linear problem with irregular data. 2021. ⟨hal-03105385⟩

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