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Minimal degree $ H(\mathrm{curl})$ and $ H(\mathrm{div})$ conforming finite elements on polytopal meshes

Authors: Wenbin Chen and Yanqiu Wang
Journal: Math. Comp. 86 (2017), 2053-2087
MSC (2010): Primary 65N30
Published electronically: October 27, 2016
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Abstract: We construct $ H(\mathrm {curl})$ and $ H(\mathrm {div})$ conforming finite elements on convex polygons and polyhedra with minimal possible degrees of freedom, i.e., the number of degrees of freedom is equal to the number of edges or faces of the polygon/polyhedron. The construction is based on generalized barycentric coordinates and the Whitney forms. In 3D, it currently requires the faces of the polyhedron be either triangles or parallelograms. Formulas for computing basis functions are given. The finite elements satisfy discrete de Rham sequences in analogy to the well-known ones on simplices. Moreover, they reproduce existing $ H(\mathrm {curl})$- $ H(\mathrm {div})$ elements on simplices, parallelograms, parallelepipeds, pyramids and triangular prisms. The approximation property of the constructed elements is also analyzed by showing that the lowest-order simplicial Nédélec-Raviart-Thomas elements are subsets of the constructed elements on arbitrary polygons and certain polyhedra.

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Additional Information

Wenbin Chen
Affiliation: Shanghai Key Laboratory for Contemporary Applied Mathematics, School of Mathematical Sciences, Fudan University, Shanghai, People’s Republic of China

Yanqiu Wang
Affiliation: Department of Mathematics, Oklahoma State University, Stillwater, Oklahoma 74074
Address at time of publication: School of Mathematical Sciences, Nanjing Normal University, Nanjing, People’s Republic of China

Keywords: $H(\mathrm{curl})$, $H(\mathrm{div})$, mixed finite element, finite element exterior calculus, generalized barycentric coordinates
Received by editor(s): February 6, 2015
Received by editor(s) in revised form: October 2, 2015, and February 23, 2016
Published electronically: October 27, 2016
Additional Notes: The first author was supported by the Key Project National Science Foundation of China (91130004) and the Natural Science Foundation of China (11671098, 11331004)
The second author was supported by the Key Laboratory of Mathematics for Nonlinear Sciences (EZH1411108/001) of Fudan University and the Ministry of Education of China and State Administration of Foreign Experts Affairs of China under the 111 project grant (B08018), and the Natural Science Foundation of China (11671210). The second author is the corresponding author.
Article copyright: © Copyright 2016 American Mathematical Society