Finite element exterior calculus

Finite element exterior calculus (FEEC) is a mathematical framework that formulates finite element methods using chain complexes. 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.

One starts with the recognition that the used differential operators are often part of complexes: successive application results in zero. Then, the phrasing of the differential operators of relevant differential equations and relevant boundary conditions as a Hodge Laplacian. The Hodge Laplacian terms are split using the Hodge decomposition. A related variational saddle-point formulation for mixed quantities is then generated. Discretization to a mesh-related subcomplex is done requiring a collection of projection operators which commute with the differential operators. One can then prove uniqueness and optimal convergence as function of mesh density.

FEEC is of immediate relevancy for diffusion, elasticity, electromagnetism, Stokes flow.

For the important de Rham complex, pertaining to the grad, curl and div operators, suitable family of elements have been generated not only for tetrahedrons, but also for other shaped elements such as bricks. Moreover, also conforming with them, prism and pyramid shaped elements have been generated. For the latter, uniquely, the shape functions are not polynomial. The quantities are 0-forms (scalars), 1-forms (gradients), 2-forms (fluxes), and 3-forms (densities).[13] Diffusion, electromagnetism, and elasticity,[14] Stokes flow,[15] general relativity, and actually all known complexes,[16] can all be phrased in terms the de Rham complex. For Navier-Stokes, there may be possibilities too.[17][18]

## 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.
13. ↑ Cockburn, Bernardo; Fu, Guosheng (2017-01-01). "A Systematic Construction of Finite Element Commuting Exact Sequences". SIAM Journal on Numerical Analysis 55 (4): 1650–1688. doi:10.1137/16M1073352. ISSN 0036-1429. https://epubs.siam.org/doi/10.1137/16M1073352.
14. ↑ Arnold, Douglas N.; Falk, Richard S.; Winther, Ragnar (2007-10-01). "Mixed finite element methods for linear elasticity with weakly imposed symmetry". Mathematics of Computation 76 (260): 1699–1724. doi:10.1090/S0025-5718-07-01998-9. Bibcode: 2007MaCom..76.1699A. https://www.ams.org/journal-getitem?pii=S0025-5718-07-01998-9.
15. ↑ Falk, Richard S.; Neilan, Michael (2013-01-01). "Stokes Complexes and the Construction of Stable Finite Elements with Pointwise Mass Conservation" (in en). SIAM Journal on Numerical Analysis 51 (2): 1308–1326. doi:10.1137/120888132. ISSN 0036-1429. http://epubs.siam.org/doi/10.1137/120888132.
16. ↑ "Finite element exterior calculus - 4 | Isaac Newton Institute for Mathematical Sciences". 5 March 2021. https://www.newton.ac.uk/seminar/20190712140015001.
17. ↑ Fang, Shizan (2020-03-01). "Nash Embedding, Shape Operator and Navier-Stokes Equation on a Riemannian Manifold" (in en). Acta Mathematicae Applicatae Sinica, English Series 36 (2): 237–252. doi:10.1007/s10255-020-0928-1. ISSN 1618-3932.
18. ↑ Samavaki, Maryam; Tuomela, Jukka (2020-02-01). "Navier–Stokes equations on Riemannian manifolds" (in en). Journal of Geometry and Physics 148. doi:10.1016/j.geomphys.2019.103543. ISSN 0393-0440. Bibcode: 2020JGP...14803543S. https://www.sciencedirect.com/science/article/pii/S0393044019302244.

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