A unified analysis of elliptic problems with various boundary conditions and their approximation
For numerical analysts, it provides a unified convergence theory that simplifies analysis across different boundary conditions and discretization schemes, but the results are largely incremental extensions of existing techniques.
This paper develops an abstract framework for analyzing numerical approximations of elliptic problems with various boundary conditions, unifying convergence analysis across multiple methods (conforming and non-conforming). The framework yields error estimates and applies to linear diffusion, Leray-Lions problems, and other models like flows in fractured media.
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{é} inequalities and the surjectivity of the divergence operator in appropriate spaces.