Fast Collocation Methods for Second Kind Integral Equations

Abstract

In this paper we develop fast collocation methods for integral equations of the second kind with weakly singular kernels. For this purpose, we construct multiscale interpolating functions and collocation functionals having vanishing moments. Moreover, we propose a truncation strategy for the coefficient matrix of the corresponding discrete system which forms a basis for fast algorithms. An optimal order of convergence of the approximate solutions obtained from the fast algorithms is proved and the computational complexity of the algorithms is estimated. The stability of the numerical method and the condition number of the truncated coefficient matrix are analyzed.