Research reports

Years: 2025 2024 2023 2022 2021 2020 2019 2018 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2004 2003 2002 2001 2000 1999 1998 1997 1996 1995 1994 1993 1992 1991

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}
}

Disclaimer
© Copyright for documents on this server remains with the authors. Copies of these documents made by electronic or mechanical means including information storage and retrieval systems, may only be employed for personal use. The administrators respectfully request that authors inform them when any paper is published to avoid copyright infringement. Note that unauthorised copying of copyright material is illegal and may lead to prosecution. Neither the administrators nor the Seminar for Applied Mathematics (SAM) accept any liability in this respect. The most recent version of a SAM report may differ in formatting and style from published journal version. Do reference the published version if possible (see SAM Publications).