The chain collocation method: A spectrally accurate calculus of forms

Journal of Computational Physics

Volumn 257, Issue PB, 2014, Pages 1352-1372

  Rufat, Dzhelil ^a   Mason, Gemma ^a   Mullen, Patrick ^a   Desbrun, Mathieu ^a  

^a CALIFORNIA INSTITUTE OF TECHNOLOGY   (United States)

Author keywords

Chains; Cochains; Dicrete Exterior Calculus; Hodge star; Spectral accuracy; Wedge product

Indexed keywords

ALGEBRA; FAST FOURIER TRANSFORMS; FLUID MECHANICS; NUMERICAL METHODS;

ALGEBRAIC STRUCTURES; COCHAIN; COLLOCATION METHOD; DICRETE EXTERIOR CALCULUS; GEOMETRIC STRUCTURE; HODGE STARS; SPECTRAL ACCURACY; STRUCTURE-PRESERVING; TOPOLOGICAL STRUCTURE; WEDGE PRODUCT;

CHAINS;

EID: 84908566119     PISSN: 00219991     EISSN: 10902716     Source Type: Journal    
DOI: 10.1016/j.jcp.2013.08.011     Document Type: Article

References (40)

1. Arnold D.N., Falk R.S., Winther R. Finite element exterior calculus, homological techniques, and applications. Acta Numer. 2006, 15:1-155.
2. Stern A., Tong Y., Desbrun M., Marsden J.E. Variational integrators for Maxwell's equations with sources. PIERS Online 2008, 4(7):711-715.
3. Arnold D.N., Falk R.S., Winther R. Finite element exterior calculus: from Hodge theory to numerical stability. Bull. Am. Math. Soc. 2010, 47:281-354.
4. Pavlov D., Mullen P., Tong Y., Kanso E., Marsden J.E., Desbrun M. Structure-preserving discretization of incompressible fluids. Physica D 2011, 240(6):443-458.
5. Holst M., Stern A. Geometric variational crimes: Hilbert complexes, finite element exterior calculus, and problems on hypersurfaces. Found. Comput. Math. 2012, 12(3):263-293.
6. Gawlik E., Mullen P., Pavlov D., Marsden J., Desbrun M. Geometric, variational discretization of continuum theories. Physica D 2011, 240(21):1724-1760.
7. Bossavit A. Computational Electromagnetism 1998, Academic Press, Boston.
8. Desbrun M., Kanso E., Tong Y. Discrete differential forms for computational modeling. Discrete Differential Geometry 2007, Springer. A. Bobenko, P. Schröder (Eds.).
9. Bochev P., Hyman J. Principles of mimetic discretizations of differential operators. IMA Vol. Math. Appl. 2006, vol. 142:89-119.
10. Grady L.J., Polimeni J.R. Discrete Calculus: Applied Analysis on Graphs for Computational Science 2010, Springer.
11. Compatible Spatial Discretizations. IMA Vol. Math. Appl. 2006, vol. 142. Springer. D.N. Arnold, P.B. Bochev, R.B. Lehoucq, R.A. Nicolaides, M. Shashkov (Eds.).
12. Nicolaides R.A., Wu X. Covolume solutions of three dimensional div-curl equations. SIAM J. Numer. Anal. 1997, 34:2195.
13. Desbrun M., Hirani A.N., Marsden J.E. Discrete exterior calculus for variational problems in computer vision and graphics. Proc. CDC 2003, vol. 42:533-538.
14. Cartan É. Les Systèmes Differentiels Exterieurs et leurs Applications Géometriques 1945, Hermann, Paris.
15. Munkres J.R. Elements of Algebraic Topology 1984, Addison-Wesley, Menlo Park, CA.
16. Burke W.L. Applied Differential Geometry 1985, Cambridge University Press.
17. Abraham R., Marsden J.E., Ratiu T. Manifolds, Tensor Analysis, and Applications. Appl. Math. Sci. 1988, vol. 75. Springer.
18. Flanders H. Differential Forms and Applications to Physical Sciences 1990, Dover Publications.
19. Frankel T. The Geometry of Physics 2004, Cambridge University Press, United Kingdom. second edition.
20. Robidoux N., Steinberg S. A discrete vector calculus in tensor grids. Comput. Methods Appl. Math. 2011, 11(1):23-66.
21. Whitney H. Geometric Integration Theory 1957, Princeton University Press.
22. Nédélec J.-C. Mixed finite elements in R3. Numer. Math. 1980, 35:315-341.
23. Tong Y., Alliez P., Cohen-Steiner D., Desbrun M. Designing quadrangulations with discrete harmonic forms. Symposium on Geometry Processing 2006, 201-210. A. Sheffer, K. Polthier (Eds.).
24. Mullen P., de Goes F., Memari P., Desbrun M. HOT: Hodge Optimized Triangulations. ACM Trans. Graph. 2011, 30(3).
25. Wang K., Weiwei, Tong Y., Desbrun M., Schröder P. Edge subdivision schemes and the construction of smooth vector fields. ACM Trans. Graph. 2006, 25(3):1041-1048.
26. Bruno O.P., Han Y., Pohlman M.M. Accurate, high-order representation of complex three-dimensional surfaces via Fourier continuation analysis. J. Comput. Phys. 2007, 227(2):1094-1125.
27. Canuto C., Hussaini M., Quarteroni A., Zang T. Spectral Methods: Fundamentals in Single Domains. Sci. Comput. Ser. 2006, Springer.
28. Kopriva D.A. A staggered-grid multidomain spectral method for the compressible Navier-Stokes equations. J. Comput. Phys. 1998, 143(1):125-158.
29. N. Robidoux, Polynomial histopolation, superconvergent degrees of freedom, and pseudospectral discrete Hodge operators, preprint, 2006.
30. Bouman M., Palha A., Kreeft J., Gerritsma M. A conservative spectral element method for curvilinear domains. Lect. Notes Comput. Sci. Eng. 2011, vol. 76:111-119. Springer, Berlin, Heidelberg. T.J. Barth (Ed.).
31. Gerritsma M. Edge functions for spectral element methods. Lect. Notes Comput. Sci. Eng. 2011, vol. 76:199-207. Springer, Berlin, Heidelberg. J.S. Hesthaven (Ed.).
32. Palha A., Gerritsma M. Spectral element approximation of the Hodge-* operator in curved elements. Lect. Notes Comput. Sci. Eng. 2011, vol. 76:283-291. Springer, Berlin, Heidelberg. J.S. Hesthaven (Ed.).
33. Crane K., Desbrun M., Schröder P. Trivial connections on discrete surfaces. Comput. Graph. Forum 2010, 29(5):1525-1533.
34. Hirani A.N. Discrete exterior calculus 2003, PhD thesis, California Institute of Technology, Pasadena, CA.
35. Leok M. Foundations of computational geometric mechanics 2004, PhD thesis, California Institute of Technology, Pasadena, CA.
36. Harrison J. Ravello lecture notes on geometric calculus-Part I arxiv:math-ph/0501001.
37. Desbrun M., Hirani A.N., Leok M., Marsden J.E. Discrete exterior calculus arxiv:math/0508341v2.
38. Kotiuga P.R. Theoretical limitations of discrete exterior calculus in the context of computational electromagnetics. IEEE Trans. Magn. 2008, 44(6):1162-1165.
39. Mullen P., McKenzie A., Pavlov D., Durant L., Tong Y., Kanso E., Marsden J.E., Desbrun M. Discrete Lie advection of differential forms. Found. Comput. Math. 2011, 11(2):131-149.
40. Trefethen L.N. Spectral Methods in MATLAB 2000, SIAM, Philadelphia.