Abstract
We consider new degrees of freedom for higher order differential forms on cubical meshes. The approach is inspired by the idea of Rapetti and Bossavit to define higher order Whitney forms and their degrees of freedom using small simplices. We show that higher order differential forms on cubical meshes can be defined analogously using small cubes and prove that these small cubes yield unisolvent degrees of freedom. Importantly, this approach is compatible with discrete exterior calculus and expands the framework to cover higher order methods on cubical meshes, complementing the earlier strategy based on simplices.
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References
Arnold, D., Logg, A.: Periodic table of the finite elements. SIAM News, 47(9), 2014
Arnold, D.N.: Spaces of finite element differential forms. In Analysis and Numerics of Partial Differential Equations, volume 4 of Springer INdAM Series, pages 117–140. Springer, 2013
Arnold, D.N., Boffi, D., Bonizzoni, F.: Finite element differential forms on curvilinear cubic meshes and their approximation properties. Numer. Math. 129(1), 1–20 (2015)
Arnold, D.N., Falk, R.S., Winther, R.: Finite element exterior calculus, homological techniques, and applications. Acta Numer 15, 1–155 (2006)
Arnold, D.N., Falk, R.S., Winther, R.: Geometric decompositions and local bases for spaces of finite element differential forms. Comput. Methods Appl. Mech. Eng. 198(21–26), 1660–1672 (2009)
Kettunen, L., Lohi, J., Räbinä, J., Mönkölä, S., Rossi, T.: Generalized finite difference schemes with higher order Whitney forms. ESAIM: Mathematical Modelling and Numerical Analysis, 55(4):1439–1460, 2021
Lohi, J.: Discrete exterior calculus and higher order Whitney forms. Master’s thesis, University of Jyväskylä, 2019
Lohi, J.: Systematic implementation of higher order Whitney forms in methods based on discrete exterior calculus. Numer. Algorithms 91(3), 1261–1285 (2022)
Lohi, J., Kettunen, L.: Whitney forms and their extensions. J. Comput. Appl. Math. 393, 113520 (2021)
Rapetti, F., Bossavit, A.: Whitney forms of higher degree. SIAM J. Numer. Anal. 47(3), 2369–2386 (2009)
Vermolen, F.J., Segal, A.: On an integration rule for products of barycentric coordinates over simplexes in \({\mathbb{R}^{n}}\). J. Comput. Appl. Math. 330, 289–294 (2018)
Whitney, H.: Geometric Integration Theory. Princeton University Press, 1957
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University of Jyväskylä. Open Access funding provided by University of Jyväskylä (JYU).
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Communicated by: Francesca Rapetti
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Lohi, J. New degrees of freedom for differential forms on cubical meshes. Adv Comput Math 49, 42 (2023). https://doi.org/10.1007/s10444-023-10047-x
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DOI: https://doi.org/10.1007/s10444-023-10047-x