Abstract

In this paper, we discuss how to efficiently evaluate and assemble general finite element variational forms on $H(\mathrm{div})$ and $H(\mathrm{curl})$. The proposed strategy relies on a decomposition of the element tensor into a precomputable reference tensor and a mesh-dependent geometry tensor. Two key points must then be considered: the appropriate mapping of basis functions from a reference element, and the orientation of geometrical entities. To address these issues, we extend here a previously presented representation theorem for affinely mapped elements to Piola-mapped elements. We also discuss a simple numbering strategy that removes the need to contend with directions of facet normals and tangents. The result is an automated, efficient, and easy-to-use implementation that allows a user to specify finite element variational forms on $H(\mathrm{div})$ and $H(\mathrm{curl})$ in close to mathematical notation.

MSC codes

  1. 65N30
  2. 68N20

Keywords

  1. mixed finite element
  2. variational form compiler
  3. Piola

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Published In

cover image SIAM Journal on Scientific Computing
SIAM Journal on Scientific Computing
Pages: 4130 - 4151
ISSN (online): 1095-7197

History

Submitted: 23 October 2008
Accepted: 9 September 2009
Published online: 20 November 2009

MSC codes

  1. 65N30
  2. 68N20

Keywords

  1. mixed finite element
  2. variational form compiler
  3. Piola

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