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Trefftz difference schemes on irregular stencils. (English) Zbl 1188.65146
Summary: The recently developed Flexible Local Approximation MEthod (FLAME) [cf. I. Tsukerman, ibid. 211, No. 2, 659–699 (2006; Zbl 1082.65108)] produces accurate difference schemes by replacing the usual Taylor expansion with Trefftz functions – local solutions of the underlying differential equation. This paper advances and casts in a general form a significant modification of FLAME proposed recently by H. Pinheiro and P. Webb [A FLAME Molecule for 3-D electromagnetic scattering, IEEE Trans. Magn. 45, No. 3, 1120–1123 (2009)]: a least-squares fit instead of the exact match of the approximate solution at the stencil nodes. As a consequence of that, FLAME schemes can now be generated on irregular stencils with the number of nodes substantially greater than the number of approximating functions.
The accuracy of the method is preserved but its robustness is improved. For demonstration, the paper presents a number of numerical examples in 2D and 3D: electrostatic (magnetostatic) particle interactions, scattering of electromagnetic (acoustic) waves, and wave propagation in a photonic crystal. The examples explore the role of the grid and stencil size, of the number of approximating functions, and of the irregularity of the stencils.

65N06 Finite difference methods for boundary value problems involving PDEs
78M20 Finite difference methods applied to problems in optics and electromagnetic theory
78A30 Electro- and magnetostatics
35Q60 PDEs in connection with optics and electromagnetic theory
78A45 Diffraction, scattering
Full Text: DOI
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