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.
This is a preview of subscription content, log in to check access.
Buy single article
Instant access to the full article PDF.
Price includes VAT for USA
Subscribe to journal
Immediate online access to all issues from 2019. Subscription will auto renew annually.
This is the net price. Taxes to be calculated in checkout.
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)
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)
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)
Arnold, D. N.: An interior penalty finite element method with discontinuous elements. SIAM Journal on Numerical Analysis 19(4), 742–760 (1982)
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)
Braess, D.: Finite Elements - Theory, Fast Solvers, and Applications in Elasticity Theory, 3rd ed. Cambridge University Press, Cambridge (2007)
Braess, D., Schöberl, J.: Equilibrated residual error estimator for edge elements. Mathematics of Computation 77(262), 651–672 (2008)
Bruening, J., Lesch, M.: Hilbert complexes. Journal of Functional Analysis 108(1), 88–132 (1992)
Carstensen, C., Merdon, C.: Estimator competition for Poisson problems. Journal of Computational Mathematics 3, 309–330 (2010)
Christiansen, S., Munthe-Kaas, H., Owren, B.: Topics in structure-preserving discretization. Acta Numerica 20, 1–119 (2011)
Christiansen, S., Winther, R.: Smoothed projections in finite element exterior calculus. Mathematics of Computation 77(262), 813–829 (2008)
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)
Christiansen, S. H.: On the linearization of Regge calculus. Numerische Mathematik 119(4), 613–640 (2011)
Christiansen, S. H., Rapetti, F.: On high order finite element spaces of differential forms. Mathematics of Computation 85(298), 517–548 (2016)
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)
Desoer, C. A., Whalen, B. H.: A note on pseudoinverses. Journal of the Society for Industrial and Applied Mathematics 11(2), 442–447 (1963)
Dodziuk, J.: Finite-difference approach to the Hodge theory of harmonic forms. American Journal of Mathematics pp. 79–104 (1976)
Falk, R. S., Winther, R.: Local bounded cochain projection. Mathematics of Computation 83(290), 2631–2656 (2014)
Gallot, S., Hulin, D., Lafontaine, J.: Riemannian Geometry, 3rd edn. Universitext. Springer-Verlag Berlin Heidelberg (2004)
Gelfand, S. I., Manin, Y. I.: Homological Algebra, Encyclopedia of Mathematical Sciences, vol. 38. Springer-Verlag Berlin Heidelberg (1999)
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)
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)
Krantz, S. G., Parks, H. R.: Geometric Integration Theory. Birkhuser, Boston, MA (2008)
Lee, J. M.: Introduction to Topological Manifolds, Graduate Texts in Mathematics, vol. 202. Springer, New York (2011)
Lee, J. M.: Introduction to Smooth Manifolds, Graduate Texts in Mathematics, vol. 218, 2nd ed. Springer, New York (2012)
MacLane, S.: Homology, Classics in Mathematics, vol. 114. Springer-Verlag Berlin Heidelberg (1995)
Osborne, M. S.: Basic Homological Algebra, Graduate Texts in Mathematics, vol. 196. Springer-Verlag, New York (2000)
Repin, S. I.: A Posteriori Estimates for Partial Differential Equations, Radon Series on Computational and Applied Mathematics, vol. 4. Walter de Gruyter, Berlin (2008)
de Rham, G.: Differentiable Manifolds: Forms, Currents, Harmonic Forms, Grundlehren der Math. Wissenschaften, vol. 266. Springer-Verlag Berlin Heidelberg (1984)
Spanier, E. H.: Algebraic Topology. Springer-Verlag, New York (1995). Corrected reprint of the 1966 original
Verfürth, R.: A Posteriori Error Estimation Techniques for Finite Element Methods. Oxford University Press, Oxford (2013)
Weil, A.: Sur les théorèmes de de Rham. Commentarii Mathematici Helvetici 26(1), 119–145 (1952)
Zaglmayr, S.: High order finite element methods for electromagnetic field computation. Universität Linz, Dissertation (2006)
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.
This research was supported by the European Research Council through the FP7-IDEAS-ERC Starting Grant scheme, project 278011 STUCCOFIELDS.
Communicated by Albert Cohen.
About this article
Cite this article
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
- Discrete distributional differential form
- Finite element exterior calculus
- Finite element method
- Harmonic form
- A posteriori error estimation
Mathematics Subject Classification