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Discrete Regular Decompositions of Tetrahedral Discrete 1-Forms

by R. Hiptmair and C. Pechstein

(Report number 2017-47)

Abstract
For a piecewise polynomial finite element space \(\boldsymbol{\mathcal{W}}^1_{p,\Gamma_D}(\mathcal{T}) \subset \boldsymbol{H}_{\Gamma_D}(\mathbf{curl}, \Omega)\) built on a mesh \(\mathcal{T}\) of a Lipschitz domain \(\Omega \subset \mathbb{R}^{3}\) and with vanishing tangential trace on \(\Gamma_D \subset \partial\Omega\), a discrete regular decomposition is a \emph{stable} splitting of elements of \(\boldsymbol{\mathcal{W}}^1_{p,\Gamma_D}(\mathcal{T})\) into (i) piecewise polynomial continuous vector fields on \(\Omega\), vanishing on \(\Gamma_D\), (ii) gradients of piecewise polynomial continuous scalar finite element functions, and (iii) a ``small'' remainder. Such decompositions have turned out to be a key tool in the numerical analysis of ``edge'' finite element methods for variational problems in \(\boldsymbol{H}_{\Gamma_D}(\mathbf{curl}, \Omega)\) that commonly occur in computational electromagnetics. We show the existence of such decompositions for N\'ed\'elec's tetrahedral edge element spaces of any polynomial degree with stability depending only on \(\Omega\), \(\Gamma_{D}\), and the shape regularity of the mesh. Our decompositions also respect homogeneous boundary conditions on a part of the boundary of \(\Omega\). Key tools for our construction are continuous regular decompositions, boundary-aware local co-chain projections, projection-based interpolation, and quasi-interpolation with low regularity requirements.

Keywords: Regular decomposition, edge elements, hp-FEM, polynomial extension, projection-based interpolation, quasi-interpolation

@Techreport{HP17_743,
  author = {R. Hiptmair and C. Pechstein},
  title = {Discrete Regular Decompositions of Tetrahedral Discrete 1-Forms},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2017-47},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2017/2017-47.pdf },
  year = {2017}
}

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