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Finite element exterior calculus with lower-order terms


Authors: Douglas N. Arnold and Lizao Li
Journal: Math. Comp. 86 (2017), 2193-2212
MSC (2010): Primary 65N30
DOI: https://doi.org/10.1090/mcom/3158
Published electronically: October 27, 2016
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Abstract: The scalar and vector Laplacians are basic operators in physics and engineering. In applications, they frequently show up 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 unify them all under the FEEC framework.


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Additional Information

Douglas N. Arnold
Affiliation: School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455
Email: arnold@umn.edu

Lizao Li
Affiliation: School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455
Email: lixx1445@umn.edu

DOI: https://doi.org/10.1090/mcom/3158
Keywords: Finite element exterior calculus, lower-order terms
Received by editor(s): September 22, 2015
Received by editor(s) in revised form: February 25, 2016
Published electronically: October 27, 2016
Additional Notes: The work of both authors was supported in part by NSF grant DMS-1418805.
Article copyright: © Copyright 2016 American Mathematical Society