Minimal degree 𝐻(𝑐𝑢𝑟𝑙) and 𝐻(𝑑𝑖𝑣) conforming finite elements on polytopal meshes

Chen, Wenbin; Wang, Yanqiu

doi:10.1090/mcom/3152

Minimal degree $H(\mathrm {curl})$ and $H(\mathrm {div})$ conforming finite elements on polytopal meshes
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by Wenbin Chen and Yanqiu Wang PDF

Math. Comp. 86 (2017), 2053-2087 Request permission

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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