Past Event: Oden Institute Seminar

The Power of Trefftz Approximations: Applications in Electromagnetic Analysis

Igor Tsukerman, Department of Electrical and Computer Engineering, The University of Akron, OH

Abstract

Trefftz approximations, by definition, involve functions that satisfy (locally) the underlying differential equation of a given problem, along with the relevant interface boundary conditions. Examples include harmonic polynomials for the Laplace equation; plane waves, cylindrical or spherical harmonics for wave problems, and so on. After a brief review of various applications of Trefftz functions, the talk will focus on two particular examples: non-asymptotic / nonlocal homogenization of periodic structures and special high-order difference schemes for electromagnetic scattering. These examples beg the question: what explains the “unreasonable effectiveness” of Trefftz approximations (to paraphrase Eugene Wigner)? In the mathematical literature, this question has been studied primarily for homogeneous media (e.g. plane wave or cylindrical wave expansions) but needs to be posed much more broadly. Bio Igor Tsukerman is Professor of Electrical and Computer Engineering at the University of Akron, Ohio, where he has been a faculty member since 1995. He has about 180 refereed publications, has authored the monograph Computational Methods for Nanoscale Applications: Particles, Plasmons and Waves (Springer 2008) and co-edited another book, Plasmonics and Plasmonic Metamaterials (World Scientific 2011). Currently he is acting as Editor-in-Chief of a five-volume reference set on electromagnetic analysis and simulation, to be published by World Scientific in 2018. He is also working on the 2nd edition of Computational Methods for Nanoscale Applications. Before coming to the University of Akron, Tsukerman was with the Department of Electrical & Computer Engineering, the University of Toronto (1990–1995). Tsukerman’s academic degrees are from St. Petersburg Polytechnic in Russia: a combined B.Sc. / M.Sc. degree (with honors) in Control Systems (1982) and a Ph.D. in Electrical Engineering (1988).