Discretization of mixed formulations of elliptic problems on polyhedral meshes

TL;DR

This paper reviews design principles for discretizing mixed formulations of elliptic problems on polyhedral meshes, establishing connections between different schemes and demonstrating the framework's flexibility for higher-order and nonlinear problems.

Contribution

It shows that consistency and stability lead to mimetic schemes across discretization frameworks and illustrates the framework's adaptability for advanced schemes.

Findings

Connections between mixed-hybrid schemes and mimetic principles

Derivation of higher-order mimetic schemes

Development of convergent schemes for nonlinear problems

Abstract

We review basic design principles underpinning the construction of mimetic finite difference and a few finite volume and finite element schemes for mixed formulations of elliptic problems. For a class of low-order mixed-hybrid schemes, we show connections between these principles and prove that the consistency and stability conditions must lead to a member of the mimetic family of schemes regardless of the selected discretization framework. Finally, we give two examples of using flexibility of the mimetic framework: derivation of higher-order schemes and convergent schemes for nonlinear problems with small diffusion coefficients.