Complexes of Discrete Distributional Differential Forms and Their Homology Theory

Abstract

Complexes of discrete distributional differential forms are introduced into finite element exterior calculus. Thus, we generalize a notion of Braess and Schöberl, originally studied for a posteriori error estimation. We construct isomorphisms between the simplicial homology groups of the triangulation, the discrete harmonic forms of the finite element complex, and the harmonic forms of the distributional finite element complexes. As an application, we prove that the complexes of finite element exterior calculus have cohomology groups isomorphic to the de Rham cohomology, including the case of partial boundary conditions. Poincaré–Friedrichs-type inequalities will be studied in a subsequent contribution.

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Acknowledgments

The author would like to thank Sören Bartels for drawing the author’s attention to [9] and supervising the diploma thesis from which this work evolved, and Snorre Christiansen for productive discussions and sharing his notes on double complexes with the author. Helpful remarks by Jeonghun Lee are appreciated.

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Correspondence to Martin Werner Licht.

Additional information

This research was supported by the European Research Council through the FP7-IDEAS-ERC Starting Grant scheme, project 278011 STUCCOFIELDS.

Communicated by Albert Cohen.

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Licht, M.W. Complexes of Discrete Distributional Differential Forms and Their Homology Theory. Found Comput Math 17, 1085–1122 (2017). https://doi.org/10.1007/s10208-016-9315-y

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Keywords

  • Discrete distributional differential form
  • Finite element exterior calculus
  • Finite element method
  • Harmonic form
  • A posteriori error estimation

Mathematics Subject Classification

  • 65N30
  • 58A12