The spline collocation method for parabolic boundary integral equations on smooth curves

Abstract

We consider the spline collocation method for a class of parabolic pseudodifferential operators. We show optimal order convergence results in a large scale of anisotropic Sobolev spaces. The results cover the classical boundary integral equations for the heat equation in the general case where the spatial domain has a smooth boundary in the plane. Our proof is based on a localization technique for which we use our recent results proved for parabolic pseudodifferential operators. For the localization we need also some special spline approximation results in anisotropic Sobolev spaces.

A semi-analytical approach to Green׳s functions for heat equation in regions of irregular shape

“…The collocation methods, on the other hand, are very explicit, quite simple in use, and attractive for users in engineering community, but their rigorous mathematical analysis is not available, leaving opened the issue of their convergence and stability. That is why this area of numerical analysis is attracting many researchers in nowadays [17,[32][33][34][35].…”

Parabolic boundary integral operators symbolic representation and basic properties

“…We need the localization result of Corollary 5.4 in the forthcoming paper [6] where we extend the spline collocation results from circular cylinders, see [5], to the case of cylinders defined by a smooth closed curve. New mapping properties related to localization are shown in Corollaries 3.t3, 4.6 and 5.4.…”

Fast Nyström Methods for Parabolic Boundary Integral Equations

“…This result is the background for the analysis of the Galerkin method in both space and time. Spline collocation methods for two dimensional domains where Fourier series techniques can be used have been discussed as well, see [21,9]. Another discretization approach is the convolution quadrature method for the time discretization [27,34].…”