Rust Implementation of Finite Element Exterior Calculus on Coordinate-Free Simplicial Complexes

TL;DR

This paper introduces a new Rust-based finite element library that leverages Finite Element Exterior Calculus to solve PDEs on coordinate-free simplicial complexes, emphasizing intrinsic metrics and eigenvalue problems.

Contribution

It presents a novel implementation of FEEC in Rust for coordinate-free complexes, including convergence verification and applications to elliptic eigenvalue problems.

Findings

Successful implementation of FEEC in Rust

Verification through convergence studies

Application to elliptic Hodge-Laplace problems

Abstract

This thesis presents the development of a novel finite element library in Rust based on the principles of Finite Element Exterior Calculus (FEEC). The library solves partial differential equations formulated using differential forms on abstract, coordinate-free simplicial complexes in arbitrary dimensions, employing an intrinsic Riemannian metric derived from edge lengths via Regge Calculus. We focus on solving elliptic Hodge-Laplace eigenvalue and source problems on the nD de Rham complex. We restrict ourselves to first-order Whitney basis functions. The implementation is partially verified through convergence studies.