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Article Dans Une Revue IMA Journal of Numerical Analysis Année : 2023

Discrete Gagliardo-Nirenberg inequality and application to the finite volume approximation of a convection-diffusion equation with a Joule effect term

Résumé

A discrete order-two Gagliardo–Nirenberg inequality is established for piecewise constant functions defined on a two-dimensional structured mesh composed of rectangular cells. As in the continuous framework, this discrete Gagliardo–Nirenberg inequality allows to control in particular the $L^4$ norm of the discrete gradient of the numerical solution by the $L^2$ norm of its discrete Hessian times its $L^\infty$ norm. This result is crucial for the convergence analysis of a finite volume method for the approximation of a convection–diffusion equation involving a Joule effect term on a uniform mesh in each direction. The convergence proof relies on compactness arguments and on a priori estimates under a smallness assumption on the data, which is essential also in the continuous framework.
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Dates et versions

hal-03881410 , version 1 (01-12-2022)
hal-03881410 , version 2 (21-12-2023)

Identifiants

Citer

Caterina Calgaro, Clément Cancès, Emmanuel Creusé. Discrete Gagliardo-Nirenberg inequality and application to the finite volume approximation of a convection-diffusion equation with a Joule effect term. IMA Journal of Numerical Analysis, 2023, ⟨10.1093/imanum/drad063⟩. ⟨hal-03881410v1⟩
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