Finite element exterior calculus

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Finite element exterior calculus (FEEC) is a mathematical framework that formulates finite element methods in the calculus of differential forms. Its main application has been a comprehensive theory for finite element methods in computational electromagnetism. FEEC has been 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[edit]

  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.
  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.