Construction of $H(\rm{div})$-Conforming Mixed Finite Elements on Cuboidal Hexahedra

TL;DR

This paper extends 2D mixed finite elements for quadrilaterals to 3D cuboidal hexahedra, providing a systematic way to define divergence-free supplemental functions with prescribed normal flux for optimal $H( m{div})$ approximation.

Contribution

It introduces a systematic procedure for constructing $H( m{div})$-conforming finite elements on cuboidal hexahedra, including divergence-free supplemental functions with prescribed normal flux.

Findings

Constructed $H( m{div})$-conforming elements on hexahedra.

Provided systematic procedures for supplemental functions.

Achieved optimal approximation spaces with minimal local dimension.

Abstract

We generalize the two dimensional mixed finite elements of Arbogast and Correa [T. Arbogast and M. R. Correa, SIAM J. Numer. Anal., 54 (2016), pp. 3332--3356] defined on quadrilaterals to three dimensional cuboidal hexahedra. The construction is similar in that polynomials are used directly on the element and supplemented with functions defined on a reference element and mapped to the hexahedron using the Piola transform. The main contribution is providing a systematic procedure for defining supplemental functions that are divergence-free and have any prescribed polynomial normal flux. General procedures are also presented for determining which supplemental normal fluxes are required to define the finite element space. Both full and reduced $H(\rm{div})$-approximation spaces may be defined, so the scalar variable, vector variable, and vector divergence are approximated optimally. The spaces can be constructed to be of minimal local dimension, if desired.