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A Posteriori Error Estimates for Finite Element Exterior Calculus: The de Rham Complex

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Abstract

Finite element exterior calculus (FEEC) has been developed over the past decade as a framework for constructing and analyzing stable and accurate numerical methods for partial differential equations by employing differential complexes. The recent work of Arnold, Falk, and Winther includes a well-developed theory of finite element methods for Hodge–Laplace problems, including a priori error estimates. In this work we focus on developing a posteriori error estimates in which the computational error is bounded by some computable functional of the discrete solution and problem data. More precisely, we prove a posteriori error estimates of a residual type for Arnold–Falk–Winther mixed finite element methods for Hodge–de Rham–Laplace problems. While a number of previous works consider a posteriori error estimation for Maxwell’s equations and mixed formulations of the scalar Laplacian, the approach we take is distinguished by a unified treatment of the various Hodge–Laplace problems arising in the de Rham complex, consistent use of the language and analytical framework of differential forms, and the development of a posteriori error estimates for harmonic forms and the effects of their approximation on the resulting numerical method for the Hodge–Laplacian.

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Acknowledgments

We would like to thank Doug Arnold for helpful discussions concerning Lemma 2 and Gantumur Tsogtgerel for helpful discussions concerning Lemma 6. The work of the first author was partially supported by National Science Foundation Grant DMS-1016094 and by a grant from the Simons Foundation (228205 to Alan Demlow). The work of the second author was partially supported by National Science Foundation Grant DMS-0645604.

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Authors and Affiliations

  1. Department of Mathematics, University of Kentucky, 715 Patterson Office Tower, Lexington, KY , 40506, USA

    Alan Demlow

  2. Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 W. Green Street, Urbana, IL , 61801, USA

    Anil N. Hirani

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  1. Alan Demlow
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  2. Anil N. Hirani
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Correspondence to Alan Demlow.

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Communicated by Douglas Arnold.

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Demlow, A., Hirani, A.N. A Posteriori Error Estimates for Finite Element Exterior Calculus: The de Rham Complex. Found Comput Math 14, 1337–1371 (2014). https://doi.org/10.1007/s10208-014-9203-2

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