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Trefftz Co-chain Calculus

by D. Casati and L. Codecasa and R. Hiptmair and F. Moro

(Report number 2019-19)

Abstract
We propose a comprehensive approach to obtain systems of equations that discretize linear stationary or time-harmonic elliptic problems in unbounded domains. This is achieved by coupling any numerical method that fits co-chain calculus with a Trefftz method. The framework of co-chain calculus accommodates both finite element exterior calculus and discrete exterior calculus. It encompasses methods based on volume meshes: its application is therefore confined to bounded domains. Conversely, Trefftz methods are based on functions that solve the homogeneous equations exactly in the unbounded complement of the meshed domain, while satisfying suitable conditions at infinity. An example of a Trefftz method is the Multiple Multipole Program (MMP), which makes use of multipoles, i.e. solutions spawned by point sources with central singularities that are placed outside the domain of approximation. In our approach the degrees of freedom describing these sources can be eliminated by computing the Schur complement of the system for the coupling, therefore leading to a boundary term for co-chain calculus that takes into account the exterior problem. As a concrete example, we specialize this general framework for the cell method, a particular variant of discrete exterior calculus, coupled with MMP to solve frequency-domain eddy-current problems. A numerical experiment shows the effectiveness of this approach.

Keywords: co-chain calculus, finite element exterior calculus, discrete exterior calculus, cell method, Trefftz method, method of auxiliary sources, multiple multipole program

@Techreport{CCHM19_823,
  author = {D. Casati and L. Codecasa and R. Hiptmair and F. Moro},
  title = {Trefftz Co-chain Calculus},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2019-19},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2019/2019-19.pdf },
  year = {2019}
}

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