Flexible approximation schemes with numerical and semi‐analytical bases

Abstract

Purpose

The purpose of this paper is to extend the generalized finite‐difference calculus of flexible local approximation methods (FLAME) to problems where local analytical solutions are unavailable.

Design/methodology/approach

FLAME uses accurate local approximations of the solution to generate difference schemes with small consistency errors. When local analytical approximations are too complicated, semi‐analytical or numerical ones can be used instead. In the paper, this strategy is applied to electrostatic multi‐particle simulations and to electromagnetic wave propagation and scattering. The FLAME basis is constructed by solving small local finite‐element problems or, alternatively, by a local multipole‐multicenter expansion. As yet another alternative, adaptive FLAME is applied to problems of wave propagation in electromagnetic (photonic) crystals.

Findings

Numerical examples demonstrate the high rate of convergence of new five‐ and nine‐point schemes in 2D and seven‐ and 19‐point schemes in 3D. The accuracy of FLAME is much higher than that of the standard FD scheme. This paves the way for solving problems with a large number of particles on relatively coarse grids. FLAME with numerical bases has particular advantages for the multi‐particle model of a random or quasi‐random medium.

Research limitations/implications

Irregular stencils produced by local refinement may adversely affect the accuracy. This drawback could be rectified by least squares FLAME, where the number of stencil nodes can be much greater than the number of basis functions, making the method more robust and less sensitive to the irregularities of the stencils.

Originality/value

Previous applications of FLAME were limited to purely analytical basis functions. The present paper shows that numerical bases can be successfully used in FLAME when analytical ones are not available.

Citation

Dai, J., Pinheiro, H., Webb, J.P. and Tsukerman, I. (2011), "Flexible approximation schemes with numerical and semi‐analytical bases", COMPEL - The international journal for computation and mathematics in electrical and electronic engineering, Vol. 30 No. 2, pp. 552-573. https://doi.org/10.1108/03321641111101078