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Three families of grad div-conforming finite elements

Numerische Mathematik volume 152, pages 701–724 (2022)Cite this article

Abstract

Several smooth finite element de Rham complexes are constructed in three-dimensional space, which yield three families of \({\text {grad}}{\text {div}}\)-conforming finite elements. The simplest element has only 8 degrees of freedom (DOFs) for a tetrahedron and 14 DOFs for a 3-rectangle. We show that these elements lead to conforming and convergent approximations to quad-div problems. As a by-product, we obtain some \({\text {grad}}{\text {div}}\)-nonconforming elements. Numerical experiments validate the correctness and efficiency of the nonconforming elements for solving the quad-div problem.

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Authors and Affiliations

  1. Department of Mathematical Sciences, Michigan Technological University, Houghton, MI, 49931, USA

    Qian Zhang

  2. Department of Mathematics, Wayne State University, Detroit, MI, 48202, USA

    Zhimin Zhang

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  1. Qian Zhang
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  2. Zhimin Zhang
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Correspondence to Zhimin Zhang.

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This work is supported in part by the National Natural Science Foundation of China Grants NSAF U1930402, NSFC 12131005, and NSFC 12101036.

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Zhang, Q., Zhang, Z. Three families of grad div-conforming finite elements. Numer. Math. 152, 701–724 (2022). https://doi.org/10.1007/s00211-022-01321-z

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  • DOI: https://doi.org/10.1007/s00211-022-01321-z

Mathematics Subject Classification

  • 65N30
  • 65N15
  • 65N12

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