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

TL;DR

This paper develops a unified abstract framework for analyzing the approximation of elliptic problems with various boundary conditions, covering multiple numerical schemes and providing convergence and error estimates.

Contribution

It introduces a novel abstract setting that unifies the convergence analysis of diverse boundary conditions and approximation methods for elliptic problems.

Findings

Unified convergence analysis for multiple boundary conditions

Error estimates applicable to various numerical schemes

Application to models like flows, elasticity, and diffusion

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.