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abstract: 'Approximations by Trefftz functions are rapidly gaining popularity in the numerical solution of boundary value problems of mathematical physics. By definition, these functions satisfy locally, in weak form, the underlying differential equations of the problem, which often results in high-order or even exponential accuracy with respect to the size of the basis set. We highlight two separate examples in applied electromagnetics and photonics: (i) homogenization of periodic structures, and (ii) numerical simulation of electromagnetic waves in slab geometries. Extensive numerical evidence and theoretical considerations show that Trefftz approximations can be applied much more broadly than is traditionally done: they are effective not only in physically homogeneous regions but also in complex inhomogeneous ones. Two mechanisms underlying the high accuracy of Trefftz approximations in such complex cases are pointed out. The first one is related to trigonometric interpolation and the second one – somewhat surprisingly – to well-posedness of random matrices.'
address:
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    Department of Electrical and Computer Engineering, The University of Akron\
    Akron, OH 44325-3904, USA\
    igor@uakron.edu
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     School of Physical & Mathematical Sciences, Nanyang Technological University\
    21 Nanyang Link, Singapore 637371\
    yidong@ntu.edu.sg
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     Department of Radiology, University of Pennsylvania, Philadelphia, PA 19104, USA\
    vmarkel@pennmedicine.upenn.edu 
