AN EFFICIENT NUMERICAL SOLUTION OF SECOND-KIND FREDHOLM INTEGRAL EQUATIONS USING THE NYSTRÖM METHOD WITH ADAPTIVE QUADRATURE RULES

Abstract

An efficient adaptive Nyström method for the solution of linear Fredholm integral equations of the second kind is presented. The method does not place quadrature nodes uniformly throughout the length of the integration interval, but rather estimates the local error of the quadrature in each subinterval and refines only those subintervals which contribute most to the overall error. A composite Gauss–Legendre pair is used to produce coarse and refined approximations; other composite methods, such as adaptive Simpson or composite Gauss–Kronrod can be applied as well. Next, the quadrature nodes and weights are used to form the Nyström linear system, which is modified as the mesh is refined. The method is based on local error estimates, global tolerance control, condition-number checks, algebraic residual testing and solution-error estimation on a dense grid, to ensure accuracy and numerical stability. The primary advantage is that the relatively few quadrature points are concentrated in regions of the kernel/solution where it is rapidly varying or less smooth, and the relatively large subintervals can be used over more smooth regions. From this error analysis, it is seen that the overall numerical error is related to the quadrature approximation as well as the amplification caused by the inverse of the discrete Fredholm operator. A reliable stopping criterion should take into account the estimated quadrature error as well as the residual of the linear system, for this reason. Computational complexity, implementation procedures, convergence conditions and reproducibility of the paper are also discussed. In general, the adaptive approach is particularly helpful in problems involving localized variations, boundary layers, oscillatory behaviour or pieces that are not necessarily smooth, where uniform quadrature rules could be overly expensive in terms of number of function evaluations.