Finite Element Exterior Calculus for Parabolic Evolution Problems On Riemannian Hypersurfaces

TL;DR

This paper extends Finite Element Exterior Calculus (FEEC) to parabolic evolution problems on Riemannian hypersurfaces, providing a unified framework with error estimates and numerical examples, generalizing previous elliptic and hyperbolic analyses.

Contribution

It develops a FEEC framework for parabolic evolution problems on Riemannian manifolds, incorporating variational crimes and establishing error estimates.

Findings

Established a priori error estimates for the extended FEEC framework.

Generalized surface finite element approximation theory to Riemannian hypersurfaces.

Presented numerical examples demonstrating the theoretical results.

Abstract

Over the last ten years, the Finite Element Exterior Calculus (FEEC) has been developed as a general framework for linear mixed variational problems, their numerical approximation by mixed methods, and their error analysis. The basic approach in FEEC, pioneered by Arnold, Falk, and Winther in two seminal articles in 2006 and 2010, interprets these problems in the setting of Hilbert complexes, leading to a more general and complete understanding. Over the last five years, the FEEC framework has been extended to a broader set of problems. One such extension, due to Holst and Stern in 2012, was to problems with variational crimes, allowing for the analysis and numerical approximation of linear and geometric elliptic partial differential equations on Riemannian manifolds of arbitrary spatial dimension. Their results substantially generalize the existing surface finite element approximation theory in several respects. In 2014, Gillette, Holst, and Zhu extended the FEEC in another direction, namely to parabolic and hyperbolic evolution systems by combining the FEEC framework for elliptic operators with classical approaches for parabolic and hyperbolic operators, by viewing solutions to the evolution problem as lying in Bochner spaces (spaces of Banach-space valued parametrized curves). Related work on developing an FEEC theory for parabolic evolution problems has also been done independently by Arnold and Chen. In this article, we extend the work of Gillette-Holst-Zhu and Arnold-Chen to evolution problems on Riemannian manifolds, through the use of framework developed by Holst and Stern for analyzing variational crimes. We establish a priori error estimates that reduce to the results from earlier work in the flat (non-criminal) setting. Some numerical examples are also presented.