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Regular Decompositions of Vector Fields - Continuous, Discrete, and Structure-Preserving

by R. Hiptmair and C. Pechstein

(Report number 2019-18)

Abstract

We elaborate so-called regular decompositions of vector fields on a three-dimensional Lipschitz domain where the field and its rotation/divergence belong to \(L^2\) and where the tangential/normal component of the field vanishes on a sufficiently smooth "Dirichlet" part of the boundary. We impose no restrictions on the topology of the domain, its boundary, or the Dirichlet boundary parts.

The field is split into a regular vector field, whose Cartesian components lie in \(H^1\) and vanish on the Dirichlet boundary, and a remainder contained in the kernel of the rotation/divergence operator. The decomposition is proved to be stable not only in the natural norms, but also with respect to the \(L^2\) norm. Besides, for special cases of mixed boundary conditions, we show the existence of \(H^1\)-regular potentials that characterize the range of the rotation and divergence operator.

We conclude with results on discrete counterparts of regular decompositions for spaces of low-order discrete differential forms on simplicial meshes. Essentially, all results for function spaces carry over, though local correction terms may be necessary. These discrete regular decompositions have become an important tool in finite element exterior calculus (FEEC) and for the construction of preconditioners.

Keywords: Regular decompositions, discrete differential forms, vector potential, edge elements, face elements

@Techreport{HP19_822,
  author = {R. Hiptmair and C. Pechstein},
  title = {Regular Decompositions of Vector Fields - Continuous, Discrete, and Structure-Preserving},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2019-18},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2019/2019-18.pdf },
  year = {2019}
}

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