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Numerische Mathematik

, Volume 142, Issue 1, pp 1–32 | Cite as

Construction of \(H({\mathrm{div}})\)-conforming mixed finite elements on cuboidal hexahedra

  • Todd ArbogastEmail author
  • Zhen Tao
Article

Abstract

We generalize the two dimensional mixed finite elements of Arbogast and Correa (SIAM J Numer Anal 54:3332–3356, 2016) 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 \(H({\mathrm{div}})\)-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.

Mathematics Subject Classification

65N30 65N12 41A10 

Supplementary material

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematics C1200University of TexasAustinUSA
  2. 2.Institute for Computational Engineering and Sciences C0200University of TexasAustinUSA

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