Numerical Experiments for Darcy Flow on a Surface Using Mixed Exterior Calculus Methods

TL;DR

This paper presents preliminary numerical experiments applying mixed finite element methods based on exterior calculus to Darcy flow on surfaces, demonstrating promising convergence results using Python implementations.

Contribution

First numerical experiments using mixed exterior calculus methods for surface Darcy flow, validating their potential effectiveness and convergence.

Findings

Qualitative convergence studies are promising.

Two numerical methods demonstrated: primal-dual DEC and finite element exterior calculus.

Python implementation with PyDEC package simplifies coding.

Abstract

There are very few results on mixed finite element methods on surfaces. A theory for the study of such methods was given recently by Holst and Stern, using a variational crimes framework in the context of finite element exterior calculus. However, we are not aware of any numerical experiments where mixed finite elements derived from discretizations of exterior calculus are used for a surface domain. This short note shows results of our preliminary experiments using mixed methods for Darcy flow (hence scalar Poisson's equation in mixed form) on surfaces. We demonstrate two numerical methods. One is derived from the primal-dual Discrete Exterior Calculus and the other from lowest order finite element exterior calculus. The programming was done in the language Python, using the PyDEC package which makes the code very short and easy to read. The qualitative convergence studies seem to be promising.