Finite element exterior calculus

Finite element exterior calculus (FEEC) is a mathematical framework that formulates finite element methods using calculus of differential forms. Its main application has been a comprehensive theory for finite element methods in computational electromagnetism, computational solid and fluid mechanics. FEEC was developed in the early 2000s by Douglas N. Arnold, Richard S. Falk and Ragnar Winther, [1] [2] [3] among others. [4] [5] [6] [7] [8] [9] [10] [11] [12] Finite element exterior calculus is sometimes called as an example of a compatible discretization technique, and bears similarities with discrete exterior calculus, although they are distinct theories.

References
  1. Arnold, Douglas N., Richard S. Falk, and Ragnar Winther. "Finite element exterior calculus, homological techniques, and applications." Acta numerica 15 (2006): 1-155.
  2. Arnold, Douglas, Richard Falk, and Ragnar Winther. "Finite element exterior calculus: from Hodge theory to numerical stability." Bulletin of the American mathematical society 47.2 (2010): 281-354.
  3. Arnold, Douglas N. (2018). Finite Element Exterior Calculus. SIAM. ISBN 978-1-611975-53-6.
  4. Alan Demlow and Anil Hirani, A posteriori error estimates for finite element exterior calculus: The de Rham complex, Found. Comput. Math. 14 (2014), 1337-1371.
  5. Christiansen, Snorre, and Ragnar Winther. "Smoothed projections in finite element exterior calculus." Mathematics of Computation 77.262 (2008): 813-829.
  6. Christiansen, Snorre, and Francesca Rapetti. "On high order finite element spaces of differential forms." Mathematics of Computation 85.298 (2016): 517-548.
  7. Holst, Michael, Adam Mihalik, and Ryan Szypowski. "Convergence and optimality of adaptive methods in the finite element exterior calculus framework." arXiv preprint arXiv:1306.1886 (2013).
  8. Holst, Michael, and Ari Stern. "Geometric variational crimes: Hilbert complexes, finite element exterior calculus, and problems on hypersurfaces." Foundations of Computational Mathematics 12.3 (2012): 263-293.
  9. Hiptmair, Ralf. "Canonical construction of finite elements." Mathematics of Computation of the American Mathematical Society 68.228 (1999): 1325-1346.
  10. Hiptmair, Ralf. "Finite elements in computational electromagnetism." Acta Numerica 11 (2002): 237-339.
  11. Kirby, Robert C. "Low-complexity finite element algorithms for the de Rham complex on simplices." SIAM Journal on Scientific Computing 36.2 (2014): A846-A868.
  12. Licht, Martin Werner. On the A Priori and A Posteriori Error Analysis in Finite Element Exterior Calculus. Diss. Dissertation, Department of Mathematics, University of Oslo, Norway, 2017.


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