2014-06-30

The Power of Trefftz Methods: Difference Schemes, Absorbing Conditions and Metamaterials

Prof. Igor Tsukerman, The University of Akron, U.S.A.

4 Jul 2014, 11:00–12:30; Location: S4|10-314

In mathematical physics and engineering, Trefftz approximations by definition involve functions that satisfy the underlying differential equation of the problem (as well as the interface boundary conditions). Examples include harmonic polynomials for the Laplace equation; plane waves, cylindrical or spherical harmonics for wave problems; exponential functions for boundary layers, and so on. Trefftz approximations are well established in the context of pseudo-spectral and domain decomposition methods, but this talk calls attention to applications that are known less well or are entirely new:

  • Finite difference-Trefftz schemes of arbitrarily high order. They are obtained by replacing the classical Taylor expansions with local Trefftz approximations.
  • Numerical and analytical absorbing boundary conditions based on Trefftz approximations.
  • Boundary-difference Trefftz methods that are analogous to boundary integral equations but are completely singularity-free.
  • Homogenization of electromagnetic and photonic metamaterials: a two-scale theory involving Trefftz approximations on both coarse and fine levels. This explains, in particular, “optical magnets” – artificial magnetism at high frequencies.

This discussion of the versatility and power of Trefftz methods is intended to stimulate their application in many other areas of applied science and engineering.

Category: CE Seminar

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Technische Universität Darmstadt

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