Implementing and Testing Exterior Calculus Discretization Techniques for PDEs

Abstract

Finite element exterior calculus and discrete exterior calculus have been actively used and developed in the last two decades. In this dissertation we first work in the context of finite element exterior calculus, studying the “trimmed serendipity” finite element family by building an implementation of them that allows for analyzing their properties in practice. Then, we apply the trimmed serendipity elements to the monodomain equation, an application problem of interest in cardiac modeling. Finally, we swap to the discrete exterior calculus context where we define and apply a fractional discrete exterior derivative. By using a Python package called Firedrake, we illustrate how to implement the trimmed serendipity finite elements. Using that code, we then give a rigorous analysis of how the trimmed serendipity finite elements compare against the tensor product finite elements on a variety of toy problems. These include a projection problem, a primal formulation of the Poisson problem, a mixed formulation of the Poisson problem, and the Maxwell cavity eigenvalue problem. After testing the elements on these toy problems, we move to applying the elements to approximating the solution of the mondomain equation. This particular equation can be changed in different ways to model how electricity flows through the heart. Depending on the exact choice of model used, it can range from a simple model that shows only the macro-scale dynamics all the way to a complicated model that incorporates micro-scale dynamics of different ionic compounds. Using this model, we test the trimmed serendipity elements in conjunction with a recent implementation of Runge-Kutta methods in Firedrake. Finally, we end with studying how fractional derivatives can be discretized in the setting of discrete exterior calculus. While the discrete exterior derivative and many other operators are well defined and commonly used in discrete exterior calculus, the fractional derivative operator is a little more difficult. The nonlocal nature of a fractional operator leads to issues of how to define distances between non-adjacent simplices in a mesh, which discrete exterior calculus does not normally do. After giving a definition for a fractional discrete exterior derivative, we analyze its properties and give a discussion on some of these difficulties.