A unified analysis of elliptic problems with various boundary conditions and their approximation
From MaRDI portal
Publication:6303588
DOI10.21136/CMJ.2019.0312-18zbMath1513.65498arXiv1806.10482WikidataQ125376647 ScholiaQ125376647MaRDI QIDQ6303588
Thierry Gallouet, Raphaèle Herbin, Jérôme Droniou, Robert Eymard
Publication date: 27 June 2018
Abstract: 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 Poincar{'e} inequalities and the surjectivity of the divergence operator in appropriate spaces.
General theory of numerical analysis in abstract spaces (65J05) Linear operator approximation theory (47A58) Numerical methods for partial differential equations, boundary value problems (65N99)
This page was built for publication: A unified analysis of elliptic problems with various boundary conditions and their approximation
