A Fast Galerkin Algorithm for Singular Integral Equations

A recent paper (Delves, 1977) described a variant of the Galerkin method for linear Fredholm integral equations of the second kind with smooth kernels, for which the total solution time using N expansion functions is θ(N2 ln N) compared with the standard Galerkin count of θ(N3). We describe here a modification of this method which retains this operations count and which is applicable to weakly singular Fredholm equations of the form where K0(x, y) is a smooth kernel and Q contains a known singularity. Particular cases treated in detail include Fredholm equations with Green's function kernels, or with kernels having logarithmic singularities; and linear Volterra equations with either regular kernels or of Abel type. The case when g(x) and/or f(x) contains a known singularity is also treated. The method described yields both a priori and a posteriori error estimates which are cheap to compute; for smooth kernels (Q = 1) it yields a modified form of the algorithm described in Delves (1977) with the advantage that the iterative scheme required to solve the equations in θ(N2) operations is rather simpler than that given there.