On low order mimetic finite difference methods

TL;DR

This paper reviews and compares two families of mimetic finite difference methods, highlighting their similarities and proposing a postprocessing technique for nodal methods based on insights from mixed-type methods.

Contribution

It provides a comparative analysis of mixed-type and nodal mimetic finite difference methods and introduces a postprocessing approach for nodal solutions.

Findings

Identifies underlying similarities between the two mimetic methods.

Defines a piecewise linear postprocessing for nodal mimetic solutions.

Establishes connections between mixed-type and nodal methods.

Abstract

These pages review two families of mimetic finite difference methods: the mixed-type methods presented in [Brezzi, Lipnikov, and Simoncini, M3AS, 2005] and the nodal methods of [Brezzi, Buffa, and Lipnikov, M2AN, 2009]. The purpose of this exercise it to highlight the similitudes underlying the construction of the two families. The comparison prompts the definition of a piecewise linear postprocessing of the nodal mimetic finite difference solution, as it was done for the mixed-type method in [Cangiani and Manzini, CMAME, 2008].