Complexes of Discrete Distributional Differential Forms and Their Homology Theory
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.
KeywordsDiscrete distributional differential form Finite element exterior calculus Finite element method Harmonic form A posteriori error estimation
Mathematics Subject Classification65N30 58A12
The author would like to thank Sören Bartels for drawing the author’s attention to  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.
- 1.Ainsworth, M., Oden, J. T.: A Posteriori Error Estimation in Finite Element Analysis, Pure and Applied Mathematics: A Wiley Series of Texts, Monographs, and Tracts, vol. 37. John Wiley & Sons, Hoboken, NY (2011)Google Scholar
- 6.Barr, M.: Acyclic Models. No. 17 in CRM Monograph Series. American Mathematical Society, Providence, RI (2002)Google Scholar
- 17.Demlow, A., Hirani, A. N.: A posteriori error estimates for finite element exterior calculus: The de Rham complex. Foundations of Computational Mathematics pp. 1–35 (2014)Google Scholar
- 19.Dodziuk, J.: Finite-difference approach to the Hodge theory of harmonic forms. American Journal of Mathematics pp. 79–104 (1976)Google Scholar
- 22.Gelfand, S. I., Manin, Y. I.: Homological Algebra, Encyclopedia of Mathematical Sciences, vol. 38. Springer-Verlag Berlin Heidelberg (1999)Google Scholar
- 27.Lee, J. M.: Introduction to Smooth Manifolds, Graduate Texts in Mathematics, vol. 218, 2nd ed. Springer, New York (2012)Google Scholar
- 28.MacLane, S.: Homology, Classics in Mathematics, vol. 114. Springer-Verlag Berlin Heidelberg (1995)Google Scholar
- 31.de Rham, G.: Differentiable Manifolds: Forms, Currents, Harmonic Forms, Grundlehren der Math. Wissenschaften, vol. 266. Springer-Verlag Berlin Heidelberg (1984)Google Scholar
- 32.Spanier, E. H.: Algebraic Topology. Springer-Verlag, New York (1995). Corrected reprint of the 1966 originalGoogle Scholar
- 35.Zaglmayr, S.: High order finite element methods for electromagnetic field computation. Universität Linz, Dissertation (2006)Google Scholar