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Coupling Finite Elements and Auxiliary Sources

by D. Casati and R. Hiptmair

(Report number 2018-04)

Abstract
We consider second-order scalar elliptic boundary value problems on unbounded domains, which model, for instance, electrostatic fields. We propose a discretization that relies on a Trefftz approximation by multipole auxiliary sources in some parts of the domain and on standard mesh-based primal Lagrangian finite elements in other parts. Several approaches are developed and, based on variational saddle point theory, rigorously analyzed to couple both discretizations across the common interface: (1) least-squares-based coupling using techniques from PDE-constrained optimization; (2) coupling through Dirichlet-to-Neumann operators; and (3) three-field variational formulation in the spirit of mortar finite element methods. We compare these approaches in a series of numerical experiments.

Keywords: Finite Element Method, Multiple Multipole Program, Method of Auxiliary Sources, Trefftz method, computational electromagnetics

BibTeX

@Techreport{CH18_758,
  author = {D. Casati and R. Hiptmair},
  title = {Coupling Finite Elements and Auxiliary Sources},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2018-04},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2018/2018-04.pdf },
  year = {2018}
}

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