nach oben

Foundations of Computational Mathematics

Erschienen in:

26.04.2016

Complexes of Discrete Distributional Differential Forms and Their Homology Theory

verfasst von: Martin Werner Licht

Erschienen in: Foundations of Computational Mathematics | Ausgabe 4/2017

Einloggen

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config

KI-gestützte Suche

Aus

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.

Vorheriger Artikel Iterative Methods Based on Soft Thresholding of Hierarchical Tensors

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 390 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Maschinenbau + Werkstoffe

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 340 Zeitschriften

aus folgenden Fachgebieten:

Bauwesen + Immobilien
Business IT + Informatik
Finance + Banking
Management + Führung
Marketing + Vertrieb
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

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)

Arnold, D. N., Falk, R., Winther, R.: Finite element exterior calculus, homological techniques, and applications. Acta Numerica 15, 1–155 (2006)MathSciNetCrossRefMATH

Arnold, D. N., Falk, R., Winther, R.: Geometric decompositions and local bases for spaces of finite element differential forms. Computer Methods in Applied Mechanics and Engineering 198(21-26), 1660–1672 (2009)MathSciNetCrossRefMATH

Arnold, D. N., Falk, R., Winther, R.: Finite element exterior calculus: from Hodge theory to numerical stability. Bulletin of the American Mathematical Society 47(2), 281–354 (2010)MathSciNetCrossRefMATH

Arnold, D. N.: An interior penalty finite element method with discontinuous elements. SIAM Journal on Numerical Analysis 19(4), 742–760 (1982)MathSciNetCrossRefMATH

Barr, M.: Acyclic Models. No. 17 in CRM Monograph Series. American Mathematical Society, Providence, RI (2002)

Bott, R., Tu, L. W.: Differential Forms in Algebraic Topology, Graduate Texts in Mathematics, vol. 82. Springer-Verlag, New York (1982)CrossRefMATH

Braess, D.: Finite Elements - Theory, Fast Solvers, and Applications in Elasticity Theory, 3rd ed. Cambridge University Press, Cambridge (2007)CrossRefMATH

Braess, D., Schöberl, J.: Equilibrated residual error estimator for edge elements. Mathematics of Computation 77(262), 651–672 (2008)MathSciNetCrossRefMATH

10.

Bruening, J., Lesch, M.: Hilbert complexes. Journal of Functional Analysis 108(1), 88–132 (1992)MathSciNetCrossRefMATH

11.

Carstensen, C., Merdon, C.: Estimator competition for Poisson problems. Journal of Computational Mathematics 3, 309–330 (2010)MathSciNetMATH

12.

Christiansen, S., Munthe-Kaas, H., Owren, B.: Topics in structure-preserving discretization. Acta Numerica 20, 1–119 (2011)MathSciNetCrossRefMATH

13.

Christiansen, S., Winther, R.: Smoothed projections in finite element exterior calculus. Mathematics of Computation 77(262), 813–829 (2008)MathSciNetCrossRefMATH

14.

Christiansen, S. H.: A characterization of second-order differential operators on finite element spaces. Mathematical Models and Methods in Applied Sciences 14(12), 1881–1892 (2004)MathSciNetCrossRefMATH

15.

Christiansen, S. H.: On the linearization of Regge calculus. Numerische Mathematik 119(4), 613–640 (2011)MathSciNetCrossRefMATH

16.

Christiansen, S. H., Rapetti, F.: On high order finite element spaces of differential forms. Mathematics of Computation 85(298), 517–548 (2016)MathSciNetCrossRefMATH

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)

18.

Desoer, C. A., Whalen, B. H.: A note on pseudoinverses. Journal of the Society for Industrial and Applied Mathematics 11(2), 442–447 (1963)MathSciNetCrossRefMATH

19.

Dodziuk, J.: Finite-difference approach to the Hodge theory of harmonic forms. American Journal of Mathematics pp. 79–104 (1976)

20.

Falk, R. S., Winther, R.: Local bounded cochain projection. Mathematics of Computation 83(290), 2631–2656 (2014)MathSciNetCrossRefMATH

21.

Gallot, S., Hulin, D., Lafontaine, J.: Riemannian Geometry, 3rd edn. Universitext. Springer-Verlag Berlin Heidelberg (2004)CrossRefMATH

22.

Gelfand, S. I., Manin, Y. I.: Homological Algebra, Encyclopedia of Mathematical Sciences, vol. 38. Springer-Verlag Berlin Heidelberg (1999)

23.

Gol’dshtein, V., Mitrea, I., Mitrea, M.: Hodge decompositions with mixed boundary conditions and applications to partial differential equations on Lipschitz manifolds. Journal of Mathematical Sciences 172(3), 347–400 (2011)MathSciNetCrossRefMATH

24.

Jakab, T., Mitrea, I., Mitrea, M.: On the regularity of differential forms satisfying mixed boundary conditions in a class of Lipschitz domains. Indiana University Mathematics Journal 58(5), 2043–2072 (2009)MathSciNetCrossRefMATH

25.

Krantz, S. G., Parks, H. R.: Geometric Integration Theory. Birkhuser, Boston, MA (2008)CrossRefMATH

26.

Lee, J. M.: Introduction to Topological Manifolds, Graduate Texts in Mathematics, vol. 202. Springer, New York (2011)CrossRef

27.

Lee, J. M.: Introduction to Smooth Manifolds, Graduate Texts in Mathematics, vol. 218, 2nd ed. Springer, New York (2012)

28.

MacLane, S.: Homology, Classics in Mathematics, vol. 114. Springer-Verlag Berlin Heidelberg (1995)

29.

Osborne, M. S.: Basic Homological Algebra, Graduate Texts in Mathematics, vol. 196. Springer-Verlag, New York (2000)CrossRef

30.

Repin, S. I.: A Posteriori Estimates for Partial Differential Equations, Radon Series on Computational and Applied Mathematics, vol. 4. Walter de Gruyter, Berlin (2008)CrossRef

31.

de Rham, G.: Differentiable Manifolds: Forms, Currents, Harmonic Forms, Grundlehren der Math. Wissenschaften, vol. 266. Springer-Verlag Berlin Heidelberg (1984)

32.

Spanier, E. H.: Algebraic Topology. Springer-Verlag, New York (1995). Corrected reprint of the 1966 original

33.

Verfürth, R.: A Posteriori Error Estimation Techniques for Finite Element Methods. Oxford University Press, Oxford (2013)CrossRefMATH

34.

Weil, A.: Sur les théorèmes de de Rham. Commentarii Mathematici Helvetici 26(1), 119–145 (1952)MathSciNetCrossRefMATH

35.

Zaglmayr, S.: High order finite element methods for electromagnetic field computation. Universität Linz, Dissertation (2006)

Titel: Complexes of Discrete Distributional Differential Forms and Their Homology Theory
verfasst von: Martin Werner Licht
Publikationsdatum: 26.04.2016
Verlag: Springer US
Erschienen in: Foundations of Computational Mathematics / Ausgabe 4/2017
Print ISSN: 1615-3375
Elektronische ISSN: 1615-3383
DOI: https://doi.org/10.1007/s10208-016-9315-y

Springer Professional

Abstract

Bitte loggen Sie sich ein, um Zugang zu Ihrer Lizenz zu erhalten.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Springer Professional "Technik"

Springer Professional "Wirtschaft"

Weitere Artikel der Ausgabe 4/2017

A Universal Algorithm for Multivariate Integration

Convergence Rates of Adaptive Methods, Besov Spaces, and Multilevel Approximation

Iterative Methods Based on Soft Thresholding of Hierarchical Tensors

Higher-Dimensional Automorphic Lie Algebras

Nodal Bases for the Serendipity Family of Finite Elements

Exact Non-reflecting Boundary Conditions Revisited: Well-Posedness and Stability

Premium Partner