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Local coderivatives and approximation of Hodge Laplace problems


Authors: Jeonghun J. Lee and Ragnar Winther
Journal: Math. Comp. 87 (2018), 2709-2735
MSC (2010): Primary 65N30
DOI: https://doi.org/10.1090/mcom/3315
Published electronically: March 26, 2018
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Abstract: The standard mixed finite element approximations of Hodge
Laplace problems associated with the de Rham complex are based on proper discrete subcomplexes. As a consequence, the exterior derivatives, which are local operators, are computed exactly. However, the approximations of the associated coderivatives are nonlocal. In fact, this nonlocal property is an inherent consequence of the mixed formulation of these methods, and can be argued to be an undesired effect of these schemes. As a consequence, it has been argued, at least in special settings, that more local methods may have improved properties. In the present paper, we construct such methods by relying on a careful balance between the choice of finite element spaces, degrees of freedom, and numerical integration rules. Furthermore, we establish key convergence estimates based on a standard approach of variational crimes.


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Additional Information

Jeonghun J. Lee
Affiliation: The Institute for Computational Engineering and Sciences, University of Texas at Austin, Austin, Texas 78712
Email: jeonghun@ices.utexas.edu

Ragnar Winther
Affiliation: Department of Mathematics, University of Oslo, 0316 Oslo, Norway
Email: rwinther@math.uio.no

DOI: https://doi.org/10.1090/mcom/3315
Keywords: Perturbed mixed methods, local constitutive laws
Received by editor(s): October 27, 2016
Received by editor(s) in revised form: May 10, 2017, and July 21, 2017
Published electronically: March 26, 2018
Additional Notes: The research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013) / ERC grant agreement 339643.
Article copyright: © Copyright 2018 American Mathematical Society

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