Fast numerical collocation solutions of integral equations
Zhongying Chen, Bin Wu, Yuesheng Xu · Communications on Pure & Applied Analysis · 2007
We present a numerical implementation of a fast multiscale collocation method for solvingFredholm integral equations of the second kind with weakly singular kernels. The general setting ofsuch a collocation method was recently developed by Chen, Micchelli and Xu. Following the generalsetting, in this paper we consider three important problems for the practical use of such collocation methods.The first problem regards the construction of concrete multiscale piecewise linear, quadratic and cubicpolynomial functions and the corresponding multiscale collocation functionals. The second problem that weaddress is the practical truncation of the coefficient matrix. We propose a block truncation strategywhich allows us to compress the matrix without computing the distances between the supports of a basis function anda collocation functional. The last problem is the fast numerical solution of the large discretelinear system resulting from the compression. We make use of the multiscale structure and the sparseness of thecoefficient matrix in developing fast solver for the linear system.Numerical examples are presented to demonstrate theaccuracy and computational speed of the method.