Skip to main content
SpringerLink
Log in

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

  • Conference paper
  • First Online:
Finite Difference Methods,Theory and Applications (FDM 2014)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9045))

Included in the following conference series:

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.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Tsukerman, I.: Computational Methods for Nanoscale Applications. Particles Plasmons and Waves. Springer, New York (2007)

    Google Scholar 

  2. Tsukerman, I.: Electromagnetic applications of a new finite-difference calculus. IEEE Trans. Magn. 41(7), 2206–2225 (2005)

    Article  Google Scholar 

  3. Tsukerman, I.: A class of difference schemes with flexible local approximation. J. Comput. Phys. 211(2), 659–699 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  4. Tsukerman, I.: Trefftz difference schemes on irregular stencils. J. Comput. Phys. 229(8), 2948–2963 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  5. Pinheiro, H., Webb, J., Tsukerman, I.: Flexible local approximation models for wave scattering in photonic crystal devices. IEEE Trans. Magn. 43(4), 1321–1324 (2007)

    Article  Google Scholar 

  6. Tsukerman, I., Čajko, F.: Photonic band structure computation using FLAME. IEEE Trans. Magn. 44(6), 1382–1385 (2008)

    Article  Google Scholar 

  7. Babuška, I., Ihlenburg, F., Paik, E.T., Sauter, S.A.: A generalized finite element method for solving the Helmholtz equation in two dimensions with minimal pollution. Comput. Meth. Appl. Mech. Eng. 128, 325–359 (1995)

    Article  MATH  Google Scholar 

  8. Saltzer, C.: Discrete potential theory for two-dimensional Laplace and Poisson difference equations. Technical report 4086, National Advisory Committee on Aeronautics (1958)

    Google Scholar 

  9. Hsiao, G., Wendland, W.L.: Boundary Integral Equations. Springer, Heidelberg (2008)

    Book  MATH  Google Scholar 

  10. Ryaben’kii, V.S.: Method of Difference Potentials and Its Applications. Springer Series in Computational Mathematics, vol. 30. Springer-Verlag, Berlin (2002)

    Book  MATH  Google Scholar 

  11. Tsynkov, S.V.: On the definition of surface potentials for finite-difference operators. J. Sci. Comput. 18, 155–189 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  12. Tsukerman, I.: A singularity-free boundary equation method for wave scattering. IEEE Trans. Antennas Propag. 59(2), 555–562 (2011)

    Article  MathSciNet  Google Scholar 

  13. Martinsson, P.: Fast multiscale methods for lattice equations. Ph.D. thesis, The University of Texas at Austin (2002)

    Google Scholar 

  14. Martinsson, P., Rodin, G.: Boundary algebraic equations for lattice problems. Proc. R. Soc. A - Math. Phys. Eng. Sci. 465(2108), 2489–2503 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  15. AlKhateeb, O., Tsukerman, I.: A boundary difference method for electromagnetic scattering problems with perfect conductors and corners. IEEE Trans. Antennas Propag. 61(10), 5117–5126 (2013)

    Article  MathSciNet  Google Scholar 

  16. Petersen, S., Farhat, C., Tezaur, R.: A space-time discontinuous Galerkin method for the solution of the wave equation in the time domain. Int. J. Numer. Meth. Eng. 78(3), 275–295 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  17. Kretzschmar, F., Schnepp, S., Tsukerman, I., Weiland, T.: Discontinuous Galerkin methods with Trefftz approximations. J. Comput. Appl. Math. 270, 211–222 (2014)

    Article  MathSciNet  Google Scholar 

  18. Egger, H., Kretzschmar, F., Schnepp, S., Tsukerman, I., Weiland, T.: Transparent boundary conditions in a Discontinuous Galerkin Trefftz method. Appl. Math. Comput. 270 (submitted, 2014). http://arxiv.org/abs/1410.1899

  19. Fezoui, L., Lanteri, S., Lohrengel, S., Piperno, S.: Convergence and stability of a discontinuous Galerkin time-domain method for the 3D heterogeneous Maxwell equations on unstructured meshes. ESAIM-Math. Model Numer. 39(6), 1149–1176 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  20. Griesmair, T., Monk, P.: Discretization of the wave equation using continuous elements in time and a hybridizable discontinuous Galerkin method in space. J. Sci. Comput. 58, 472–498 (2014)

    Article  MathSciNet  Google Scholar 

  21. Lilienthal, M., Schnepp, S., Weiland, T.: Non-dissipative space-time hp -discontinuous Galerkin method for the time-dependent maxwell equations. J. Comput. Phys. 275, 589–607 (2014)

    Article  MathSciNet  Google Scholar 

  22. Moiola, A., Hiptmair, R., Perugia, I.: Plane wave approximation of homogeneous Helmholtz solutions. Z. Angew. Math. Phys. 62(5), 809–837 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  23. Hiptmair, R., Moiola, A., Perugia, I.: Plane wave discontinuous Galerkin methods for the 2D Helmholtz equation: analysis of the p-version. SIAM J. Numer. Anal. 49(1), 264–284 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  24. Huber, M., Schöberl, J., Sinwel, A., Zaglmayr, S.: Simulation of diffraction in periodic media with a coupled finite element and plane wave approach. SIAM J. Sci. Comput. 31, 1500–1517 (2009)

    Article  MATH  Google Scholar 

  25. Berenger, J.P.: A perfectly matched layer for the absorption of electromagnetic waves. J. Comput. Phys. 127, 363–379 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  26. Teixeira, F.L., Chew, W.C.: General closed-form PML constitutive tensors to match arbitrary bianisotropic and dispersive linear media. IEEE Microwave Guided Wave Lett. 8, 223–225 (1998)

    Article  Google Scholar 

  27. Sacks, Z., Kingsland, D., Lee, R., Lee, J.F.: A perfectly matched anisotropic absorber for use as an absorbing boundary condition. IEEE Trans. Antennas Propag. 43(12), 1460–1463 (1995)

    Article  Google Scholar 

  28. Gedney, S.D.: An anisotropic perfectly matched layer-absorbing medium for the truncation of FDTD lattices. IEEE Trans. Antennas Propag. 44(12), 1630–1639 (1996)

    Article  Google Scholar 

  29. Collino, F., Monk, P.B.: Optimizing the perfectly matched layer. Comput. Meth. Appl. Mech. Eng. 164, 157–171 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  30. Higdon, R.L.: Absorbing boundary conditions for difference approximations to the multidimensional wave equation. Math. Comput. 47(176), 437–459 (1986)

    MATH  MathSciNet  Google Scholar 

  31. Higdon, R.L.: Numerical absorbing boundary conditions for the wave equation. Math. Comput. 49(179), 65–90 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  32. Bayliss, A., Turkel, E.: Radiation boundary-conditions for wave-like equations. Commun Pure Appl. Math. 33(6), 707–725 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  33. Bayliss, A., Gunzburger, M., Turkel, E.: Boundary conditions for the numerical solution of elliptic equations in exterior regions. SIAM J. Appl. Math. 42(2), 430–451 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  34. Hagstrom, T., Hariharan, S.I.: A formulation of asymptotic and exact boundary conditions using local operators. Appl. Numer. Math. 27(4), 403–416 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  35. Givoli, D.: High-order nonreflecting boundary conditions without high-order derivatives. J. Comput. Phys. 170(2), 849–870 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  36. Givoli, D., Neta, B.: High-order nonreflecting boundary conditions for the dispersive shallow water equations. J. Comput. Appl. Math. 158(1), 49–60 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  37. Hagstrom, T., Warburton, T.: A new auxiliary variable formulation of high-order local radiation boundary conditions: corner compatibility conditions and extensions to first-order systems. Wave Motion 39(4), 327–338 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  38. Zarmi, A., Turkel, E.: A general approach for high order absorbing boundary conditions for the Helmholtz equation. J. Comput. Phys. 242, 387–404 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  39. Givoli, D.: High-order local non-reflecting boundary conditions: a review. Wave Motion 39(4), 319–326 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  40. Tsynkov, S.V.: Numerical solution of problems on unbounded domains. Rev. Appl. Numer. Math. 27, 465–532 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  41. Hagstrom, T.: Radiation boundary conditions for the numerical simulation of waves. In: Iserlis, A. (ed.) Acta Numerica, vol. 8, pp. 47–106. Cambridge University Press, Cambridge (1999)

    Google Scholar 

  42. Gratkowski, S.: Asymptotyczne warunki brzegowe dla stacjonarnych zagadnień elektromagnetycznych w obszarach nieograniczonych - algorytmy metody elementów skończonych. Wydawnictwo Uczelniane Zachodniopomorskiego Uniwersytetu Technologicznego (2009)

    Google Scholar 

  43. Tsukerman, I.: A “Trefftz machine" for absorbing boundary conditions. Ann. Stat. 42(3), 1070–1101 (2014). http://arxiv.org/abs/1406.0224

    Article  Google Scholar 

  44. Paganini, A., Scarabosio, L., Hiptmair, R., Tsukerman, I.: tz approximations: a new framework for nonreflecting boundary conditions (in preparation, 2015)

    Google Scholar 

  45. Soukoulis, C.M., Wegener, M.: Past achievements and future challenges in the development of three-dimensional photonic metamaterials. Nat. Photonics 5, 523–530 (2011)

    Google Scholar 

  46. Bensoussan, A., Lions, J., Papanicolaou, G.: Asymptotic Methods in Periodic Media. Elsevier, North Holland (1978)

    Google Scholar 

  47. Bakhvalov, N.S., Panasenko, G.: Homogenisation: Averaging Processes in Periodic Media. Mathematical Problems in the Mechanics of Composite Materials. Springer, The Netherlands (1989)

    Book  MATH  Google Scholar 

  48. Milton, G.: The Theory of Composites. Cambridge University Press, Cambridge; New York (2002)

    Book  MATH  Google Scholar 

  49. Bossavit, A., Griso, G., Miara, B.: Modelling of periodic electromagnetic structures bianisotropic materials with memory effects. J. Math. Pures Appl. 84(7), 819–850 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  50. Tsukerman, I.: Negative refraction and the minimum lattice cell size. J. Opt. Soc. Am. B 25, 927–936 (2008)

    Article  MathSciNet  Google Scholar 

  51. Tsukerman, I., Markel, V.A.: A nonasymptotic homogenization theory for periodic electromagnetic structures. Proc. Royal Soc. A 470 2014.0245 (2014)

    Google Scholar 

  52. Tsukerman, I.: Effective parameters of metamaterials: a rigorous homogenization theory via Whitney interpolation. J. Opt. Soc. Am. B 28(3), 577–586 (2011)

    Article  Google Scholar 

  53. Pors, A., Tsukerman, I., Bozhevolnyi, S.I.: Effective constitutive parameters of plasmonic metamaterials: homogenization by dual field interpolation. Phys. Rev. E 84, 016609 (2011)

    Article  Google Scholar 

  54. Xiong, X.Y., Jiang, L.J., Markel, V.A., Tsukerman, I.: Surface waves in three-dimensional electromagnetic composites and their effect on homogenization. Opt. Express 21(9), 10412–10421 (2013)

    Article  Google Scholar 

Download references

Acknowledgment

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.

Author information

Authors and Affiliations

  1. Graduate School of Computational Engineering, Technische Universität Darmstadt, Dolivostrasse 15, Darmstadt, Germany

    Fritz Kretzschmar

  2. Institut für Theorie Elektromagnetischer Felder, Technische Universitaet Darmstadt, Schlossgartenstrasse 8, Darmstadt, Germany

    Fritz Kretzschmar

  3. Institut Für Geophysik, ETH Zürich, Sonneggstrasse 5, 8092, Zürich, Switzerland

    Sascha M. Schnepp

  4. Department of Electrical and Computer Engineering, The University of Akron, Akron, OH, 44325-3904, USA

    Farzad Ahmadi, Nabil Nowak & Igor Tsukerman

  5. Graduate Group in Applied Mathematics and Computational Science, and Department of Radiology, Department of Bioengineering, University of Pennsylvania, Philadelphia, PA, 19104, USA

    Vadim A. Markel

  6. Department of Mathematics, TU Darmstadt, Dolivostrasse 15, Darmstadt, Germany

    Herbert Egger

Authors
  1. Fritz Kretzschmar
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Sascha M. Schnepp
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Herbert Egger
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Farzad Ahmadi
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Nabil Nowak
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. Vadim A. Markel
    View author publications

    You can also search for this author in PubMed Google Scholar

  7. Igor Tsukerman
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Igor Tsukerman .

Editor information

Editors and Affiliations

  1. Bulgarian Academy of Sciences, Institute of Information and Communication Technologies, Sofia, Bulgaria

    Ivan Dimov

  2. Institue of Mathematics and MTA-ELTE Numerical Analysis and Large Networks Research Group, Eötvös Loránd University, Budapest, Hungary

    István Faragó

  3. University of Rousse, Rousse, Bulgaria

    Lubin Vulkov

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer International Publishing Switzerland

About this paper

Cite this paper

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. https://doi.org/10.1007/978-3-319-20239-6_5

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-20239-6_5

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-20238-9

  • Online ISBN: 978-3-319-20239-6

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation