Abstract
Theses notes present the duality of Whitney forms as a tool to describe manifolds by chains and fields by cochains. Relying on this duality, we can construct compatible discretization methods for PDEs, that are methods which respect the nature of the fields involved in the equations (the degrees of freedom have a physical meaning) as well as the geometric and topological structure of the continuous model. We briefly recall how it is possible to define high-order Whitney forms just refining the chains that describe the manifolds.
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References
Arnold, D.N., Falk, R., Winther, R.: Finite element exterior calculus, homological techniques, and applications. Acta Numerica, 1–155 (2006)
Bossavit, A., Vérité, J.C.: A Mixed FEM-BIEM Method to Solve Eddy-Current Problems. IEEE Trans. on Magn. 18, 431–435 (1982)
Bossavit, A.: Computational Electromagnetism. Academic Press, New York (1998)
Bossavit, A.: Generating Whitney forms of polynomial degree one and higher. IEEE Trans. on Magn. 38, 341–344 (2002)
Burke, W.L.: Applied differential geometry. Cambridge Univ. Press, Cambridge U.K. (1985)
Christiansen, S.H., Rapetti, F.: On high order finite element spaces of differential forms. Preprint (2013)
Gerritsma, M., Hiemstra, R., Kreeft, J., Palha, A., Rebelo, P., Toshniwal, D.: The geometric basis of mimetic spectral approximations. Icosahom 2012 procs, to appear (2012)
Kotiuga, P.R.: Hodge Decompositions and Computational Electromagnetics. PhD Thesis, Dept. of Electrical Engineering, McGill University, Montréal (1984)
Nédélec, J.-C.: Mixed finite elements in R 3. Numerische Mathematik 35, 315–341 (1980)
Rapetti, F.: Weight computation for simplicial Whitney forms of degree one. C. R. Acad. Sci. Paris Ser. I 341(8), 519–523 (2005)
Rapetti, F.: High order edge elements on simplicial meshes. Meth. Math. en Anal. Num. 41(6), 1001–1020 (2007)
Rapetti, F., Bossavit, A.: Whitney forms of higher degree. SIAM J. Numer. Anal. 47(3), 2369–2386 (2009)
Whitney, H.: Geometric integration theory. Princeton Univ. Press (1957).
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Bossavit, A., Rapetti, F. (2014). Whitney Forms, from Manifolds to Fields. In: Azaïez, M., El Fekih, H., Hesthaven, J. (eds) Spectral and High Order Methods for Partial Differential Equations - ICOSAHOM 2012. Lecture Notes in Computational Science and Engineering, vol 95. Springer, Cham. https://doi.org/10.1007/978-3-319-01601-6_14
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DOI: https://doi.org/10.1007/978-3-319-01601-6_14
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