The Power of Trefftz Approximations: Finite Difference, Boundary Difference and Discontinuous Galerkin Methods; Nonreflecting Conditions and Non-Asymptotic Homogenization

  • Fritz Kretzschmar
  • Sascha M. Schnepp
  • Herbert Egger
  • Farzad Ahmadi
  • Nabil Nowak
  • Vadim A. Markel
  • Igor TsukermanEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9045)


In problems of mathematical physics, Trefftz approximations by definition involve functions that satisfy the differential equation of the problem. The power and versatility of such approximations is illustrated with an overview of a number of application areas: (i) finite difference Trefftz schemes of arbitrarily high order; (ii) boundary difference Trefftz methods analogous to boundary integral equations but completely singularity-free; (iii) Discontinuous Galerkin (DG) Trefftz methods for Maxwell’s electrodynamics; (iv) numerical and analytical nonreflecting Trefftz boundary conditions; (v) non-asymptotic homogenization of electromagnetic and photonic metamaterials.


Trefftz functions Finite difference schemes Boundary difference schemes Maxwell equations Wave propagation Effective medium theory Discontinuous galerkin methods Nonreflecting boundary conditions Metamaterials 



IT thanks Prof. Ralf Hiptmair (ETH Zürich) for very helpful discussions and, in particular, for suggesting additional discrete dof on the absorbing boundary.

The work was supported in part by the following grants: German Research Foundation (DFG) GSC 233 (FK and HE); US National Science Foundation DMS-1216970 (IT and VAM); U.S. Army Research Office W911NF1110384 (IT). SMS acknowledges support by the Alexander von Humboldt-Foundation through a Feodor-Lynen research fellowship.


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Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Fritz Kretzschmar
    • 1
    • 2
  • Sascha M. Schnepp
    • 3
  • Herbert Egger
    • 6
  • Farzad Ahmadi
    • 4
  • Nabil Nowak
    • 4
  • Vadim A. Markel
    • 5
  • Igor Tsukerman
    • 4
    Email author
  1. 1.Graduate School of Computational EngineeringTechnische Universität DarmstadtDarmstadtGermany
  2. 2.Institut für Theorie Elektromagnetischer FelderTechnische Universitaet DarmstadtDarmstadtGermany
  3. 3.Institut Für GeophysikETH ZürichZürichSwitzerland
  4. 4.Department of Electrical and Computer EngineeringThe University of AkronAkronUSA
  5. 5.Graduate Group in Applied Mathematics and Computational Science, and Department of Radiology, Department of BioengineeringUniversity of PennsylvaniaPhiladelphiaUSA
  6. 6.Department of MathematicsTU DarmstadtDarmstadtGermany

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