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 Tsukerman
Conference paper

DOI: 10.1007/978-3-319-20239-6_5

Part of the Lecture Notes in Computer Science book series (LNCS, volume 9045)
Cite this paper as:
Kretzschmar F. et al. (2015) The Power of Trefftz Approximations: Finite Difference, Boundary Difference and Discontinuous Galerkin Methods; Nonreflecting Conditions and Non-Asymptotic Homogenization. In: Dimov I., Faragó I., Vulkov L. (eds) Finite Difference Methods,Theory and Applications. FDM 2014. Lecture Notes in Computer Science, vol 9045. Springer, Cham

Abstract

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.

Keywords

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

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
  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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