The fast Fourier transform and the numerical solution of one-dimensional boundary integral equations

Abstract

Summary.Here we present a fully discretized projection method with Fourier series which is based on a modification of the fast Fourier transform. The method is applied to systems of integro-differential equations with the Cauchy kernel, boundary integral equations from the boundary element method and, more generally, to certain elliptic pseudodifferential equations on closed smooth curves. We use Gaussian quadratures on families of equidistant partitions combined with the fast Fourier transform. This yields an extremely accurate and fast numerical scheme. We present complete asymptotic error estimates including the quadrature errors. These are quasioptimal and of exponential order for analytic data. Numerical experiments for a scattering problem, the clamped plate and plane estatostatics confirm the theoretical convergence rates and show high accuracy.

A spectral Galerkin method for a boundary integral equation

“…292, 501]. Also, in a recent paper, Lamp, Schleicher, and Wendland [9] have independently studied a generalization of the method, which allows one to deal with a class of periodic, elliptic pseudodifferential equations, and thus have obtained some of our results as special cases. (They consider the spaces Hp only for p = 2.)…”

The Method of Fundamental Solutions for Stationary Heat Conduction Problems in Rotationally Symmetric Domains

“…We took M = N = 12, 16,24,32,48. From this figure we observe that, for both examples, as ε varies on the interval (10 −2 , 1), the error decreases exponentially fast.…”

“…The circulant and/or block circulant structure of the matrices appearing when the MFS is applied to axisymmetric problems has been exploited in [8,22]. The idea of exploiting the FFTs to solve axisymmetric integral equations was introduced in [12]; see also [16]. The circulant structure of the coefficient matrices also arises from the application of other boundary methods such as the boundary element method (BEM), and this has been exploited in the past (see, e.g., [3,15]).…”

Numerical solution of boundary integral equations by means of attenuation factors

“…The method in Berrut (1976) merely uses one-dimensional FFTs and solves the resulting systems of equations by iteration (see also Berrut, 1986). Many closely related methods have been suggested since (Hoidn, 1983;Arnold, 1983;Lamp et al, 1985;Atkinson, 1988). A theoretical study of the method for C ∞ -curves was performed in McLean (1986) and McLean et al (1989).…”