A unified analysis of elliptic problems with various boundary conditions and their approximation

Jérôme Droniou (1) , Robert Eymard (2) , Thierry Gallouët (3) , Raphaèle Herbin (3)

1 School of Mathematical Sciences [Clayton]
2 LAMA - Laboratoire d'Analyse et de Mathématiques Appliquées
3 I2M - Institut de Mathématiques de Marseille

We design an abstract setting for the approximation in Banach spaces of operators acting in duality. A typical example are the gradient and divergence operators in Lebesgue–Sobolev spaces on a bounded domain. We apply this abstract setting to the numerical approximation of Leray-Lions type problems, which include in particular linear diffusion. The main interest of the abstract setting is to provide a unified convergence analysis that simultaneously covers (i) all usual boundary conditions, (ii) several approximation methods. The considered approximations can be conforming, or not (that is, the approximation functions can belong to the energy space of the problem, or not), and include classical as well as recent numerical schemes. Convergence results and error estimates are given. We finally briefly show how the abstract setting can also be applied to other models, including flows in fractured medium, elasticity equations and diffusion equations on manifolds. A by-product of the analysis is an apparently novel result on the equivalence between general Poincaré inequalities and the surjectivity of the divergence operator in appropriate spaces.

Mots clés: elliptic problem; various boundary conditions; gradient discretisation method; Leray-Lions problem

Domaines: Analyse numérique [math.NA]

Jérôme Droniou, Robert Eymard, Thierry Gallouët, Raphaèle Herbin. A unified analysis of elliptic problems with various boundary conditions and their approximation. 2019. ⟨hal-01823265v2⟩