Complexes of Discrete Distributional Differential Forms and Their Homology Theory

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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Foundations of Computational Mathematics Springer Journals

Complexes of Discrete Distributional Differential Forms and Their Homology Theory

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Publisher
Springer US
Copyright
Copyright © 2016 by SFoCM
Subject
Mathematics; Numerical Analysis; Economics, general; Applications of Mathematics; Linear and Multilinear Algebras, Matrix Theory; Math Applications in Computer Science; Computer Science, general
ISSN
1615-3375
eISSN
1615-3383
D.O.I.
10.1007/s10208-016-9315-y
Publisher site
See Article on Publisher Site

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.

Journal

Foundations of Computational MathematicsSpringer Journals

Published: Apr 26, 2016

References

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