Abstract
Discrete exterior calculus (DEC) is a framework for constructing discrete versions of exterior differential calculus objects, and is widely used in computer graphics, computational topology, and discretizations of the Hodge–Laplace operator and other related partial differential equations. However, a rigorous convergence analysis of DEC has always been lacking; as far as we are aware, the only convergence proof of DEC so far appeared is for the scalar Poisson problem in two dimensions, and it is based on reinterpreting the discretization as a finite element method. Moreover, even in two dimensions, there have been some puzzling numerical experiments reported in the literature, apparently suggesting that there is convergence without consistency. In this paper, we develop a general independent framework for analyzing issues such as convergence of DEC without relying on theories of other discretization methods, and demonstrate its usefulness by establishing convergence results for DEC beyond the Poisson problem in two dimensions. Namely, we prove that DEC solutions to the scalar Poisson problem in arbitrary dimensions converge pointwise to the exact solution at least linearly with respect to the mesh size. We illustrate the findings by various numerical experiments, which show that the convergence is in fact of second order when the solution is sufficiently regular. The problems of explaining the second order convergence, and of proving convergence for general p-forms remain open.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Arnold, D.N., Falk, R.S., Winther, R.: Finite element exterior calculus, homological techniques, and applications. Acta Numerica 15, 1–155 (2006)
Bell, N., Hirani, A.N.: PyDEC: software and algorithms for discretization of exterior calculus. ACM Trans. Math. Softw. 39(1), 41 (2012)
Ciarlet, P.G.: Finite Element Method for Elliptic Problems. Society for Industrial and Applied Mathematics, Philadelphia, PA (2002)
Crane, K., de Goes, F., Desbrun, M., Schroder, P.: Digital Geometry Processing with Discrete Exterior Calculus. In: Proceedings of ACM SIGGRAPH 2013 Courses. SIGGRAPH ’13. ACM, New York, NY, USA (2013)
Desbrun, M., Hirani, A.N., Leok, M., Marsden, J.E.: Discrete Exterior Calculus. arXiv:math/0508341 (2005)
Desbrun, M., Kanso, E., Tong, Y.: Discrete Differential Forms for Computational Modeling. In: Bobenko, A.I., Sullivan, J.M., Schröder, P., Ziegler, G.M. (eds.) Discrete Differential Geometry, pp. 287–324. Birkhäuser, Basel (2008)
Frauendiener, J.: Discrete differential forms in general relativity. Class. Quantum Gravity 23(16), S369 (2006)
Hildebrandt, K., Polthier, K., Wardetzky, M.: On the convergence of metric and geometric properties of polyhedral surfaces. Geom. Dedic. 123, 89–112 (2006)
Hiptmair, R.: Discrete Hodge operators. Num. Math. 90(2), 265–289 (2001)
Hirani, A.N.: Discrete exterior calculus. Thesis (Ph.D.), California Institute of Technology. ProQuest LLC, Ann Arbor, MI, p. 103 (2003)
Hirani, A.N., Kalyanaraman, K., VanderZee, E.B.: Delaunay Hodge star. Comput. Aided Des. 45(2), 540–544 (2013)
Hirani, A.N., Nakshatrala, K.B., Chaudhry, J.H.: Numerical method for darcy flow derived using discrete exterior calculus. Int. J. Comput. Methods Eng. Sci. Mech. 16(3), 151–169 (2015)
Mohamed, M.S., Hirani, A.N., Samtaney, R.: Discrete exterior calculus discretization of incompressible Navier–Stokes equations over surface simplicial meshes. J. Comput. Phys. 312, 175–191 (2016)
Nong, H.D.: Numerical Study of Discrete Laplace–Beltrami. Tech. Rep. California Institute of Technology (2004)
Seslija, M., van der Schaft, A., Scherpen, J.M.A.: Discrete exterior geometry approach to structure-preserving discretization of distributed-parameter port-Hamiltonian systems. J. Geom. Phys. 62(6), 1509–1531 (2012)
Stern, A., Tong, Y., Desbrun, M., Marsden, J.E.: Geometric computational electrodynamics with variational integrators and discrete differential forms. Fields Inst. Commun. 73, 437–475 (2007)
Stern, A., Tong, Y., Desbrun, M., Marsden, J.E.: Variational integrators for Maxwell’s equations with sources. arXiv:0803.2070 (2008)
VanderZee, E., Hirani, A.N., Guoy, D.: Triangulation of simple 3D shapes with well-centered tetrahedra. In: Rao, V.G. (ed.). Proceedings of the 17th International Meshing Roundtable, pp. 19–35. Springer, Berlin (2008)
VanderZee, E., Hirani, A.N, Guoy, D., Ramos, E.: A.: Well-centered triangulation. SIAM J. Sci. Comput. 31(6), 4497–4523 (2009/2010)
VanderZee, E., Hirani, A.N., Zharnitsky, V., Guoy, D.: A dihedral acute triangulation of the cube. Comput. Geom. 43(5), 445–452 (2010)
VanderZee, E., Hirani, A.N., Guoy, D., Zharnitsky, V., Ramos, E.A.: Geometric and combinatorial properties of well-centered triangulations in three and higher dimensions. Comput. Geom. 46(6), 700–724 (2013)
Wardetzky, M.: Convergence of the Cotangent Formula: An Overview. Discrete Differential Geometry, pp. 275–286. Oberwolfach Semin, Birkhäuser, Basel (2008)
Whitney, H.: Geometric Integration Theory. Princeton University Press, Princeton (1957)
Xu, G.: Convergence of discrete Laplace–Beltrami operators over surfaces. Comput. Math. Appl. 48(3–4), 347–360 (2004)
Xu, G.: Discrete Laplace–Beltrami operators and their convergence. Comput. Aided Geom. Des. 21(8), 767–784 (2004)
Yavari, A.: On geometric discretization of elasticity. J. Math. Phys. 49(2), 022901 (2008)
Acknowledgements
This paper is the first author’s first research article, and he would like to thank his family for its unwavering support on his journey to become a scientist. The second author would like to thank Gerard Awanou (UI Chicago) for his insights offered during the initial stage of this project, and Alan Demlow (TAMU) for a discussion on interpreting some of the numerical experiments. This work is supported by NSERC Discovery Grant, and NSERC Discovery Accelerator Supplement Program. During the course of this work, the second author was also partially supported by the Mongolian Higher Education Reform Project L2766-MON.
Author information
Authors and Affiliations
Corresponding author
Additional information
Editor in Charge: Kenneth Clarkson
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Schulz, E., Tsogtgerel, G. Convergence of Discrete Exterior Calculus Approximations for Poisson Problems. Discrete Comput Geom 63, 346–376 (2020). https://doi.org/10.1007/s00454-019-00159-x
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00454-019-00159-x