Dual Formulations of Mixed Finite Element Methods with Applications

TL;DR

This paper analyzes the role of discrete Hodge stars in mixed finite element methods, demonstrating their impact on stability and introducing new dual mesh-based approaches for solving PDEs.

Contribution

It provides a theoretical framework linking discrete Hodge stars to stability and proposes novel mixed methods using dual meshes and interpolation functions.

Findings

Discrete Hodge star choice affects numerical stability

New mixed methods on dual meshes are introduced

Applications to magnetostatics and Darcy flow are demonstrated

Abstract

Mixed finite element methods solve a PDE using two or more variables. The theory of Discrete Exterior Calculus explains why the degrees of freedom associated to the different variables should be stored on both primal and dual domain meshes with a discrete Hodge star used to transfer information between the meshes. We show through analysis and examples that the choice of discrete Hodge star is essential to the numerical stability of the method. Additionally, we define interpolation functions and discrete Hodge stars on dual meshes which can be used to create previously unconsidered mixed methods. Examples from magnetostatics and Darcy flow are examined in detail.