20100324
Traditional finitedifference methods are typically based on Taylor expansions of the differential operator. The generalized finitedifference calculus of Flexible Local Approximation MEthods (FLAME) often dramatically improves the accuracy by replacing these Taylor expansions with Trefftz functions – local solutions of the underlying differential equation.
The principles and applications of FLAME are reviewed in the first part of the talk. For example, in the computation of Bloch bands FLAME yields 6−8 digits in the (generally complex) Bloch wavenumber on Cartesian grids of very modest size. Other numerical examples in 2D and 3D include electrostatic and magnetostatic particle interactions, scattering of electromagnetic waves, and wave propagation in a photonic crystal. A new version of FLAME that works on irregular and even random stencils is also presented.
In the second part of the talk, FLAME is used to enhance boundary difference methods that are, in contrast with the traditional boundary integral methods, singularityfree. Instead of converting Maxwell's system into an integral boundary form first and discretizing second, here the differential equations are first discretized on a regular grid and then converted to boundary difference equations. The procedure involves nonsingular Green's functions on a lattice rather than their singular continuous counterparts. Numerical examples demonstrate the effectiveness, accuracy and convergence of the method.
Category: CE SeminarTechnische Universität Darmstadt
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