Finite element exterior calculus with lower-order terms

TL;DR

This paper extends the Finite Element Exterior Calculus framework to analyze the impact of lower-order terms on mixed finite element methods for elliptic operators, revealing stability and convergence insights.

Contribution

It systematically analyzes lower-order terms in mixed methods within FEEC, proving stability and deriving improved error estimates, especially for the vector Laplacian.

Findings

Stable discretization persists with lower-order terms at fine mesh resolutions.

Degradation of convergence rates occurs for certain elements with lower-order terms.

New error estimates for vector Laplacian problems are established.

Abstract

The scalar and vector Laplacians are basic operators in physics and engineering. In applications, they show up frequently perturbed by lower-order terms. The effect of such perturbations on mixed finite element methods in the scalar case is well-understood, but that in the vector case is not. In this paper, we first show that surprisingly for certain elements there is degradation of the convergence rates with certain lower-order terms even when both the solution and the data are smooth. We then give a systematic analysis of lower-order terms in mixed methods by extending the Finite Element Exterior Calculus (FEEC) framework, which contains the scalar, vector Laplacian, and many other elliptic operators as special cases. We prove that stable mixed discretization remains stable with lower-order terms for sufficiently fine discretization. Moreover, we derive sharp improved error estimates for each individual variable. In particular, this yields new results for the vector Laplacian problem which are useful in applications such as electromagnetism and acoustics modeling. Further our results imply many previous results for the scalar problem and thus unifies them all under the FEEC framework.