Spectrally accurate Nyström-solver error bounds for 1-D Fredholm integral equations of the second kind

摘要

We present the theory underlying and computational implementation of analytical predictions of error bounds for the approximate solution of one-dimensional Fredholm integral equations of the second kind. Through asymptotic estimates of near-supremal operator norms, readily implementable formulae for the error bounds are computed explicitly using only the numerical solution of Nyström-based methods on distributions of nodes at the roots or extrema of diverse orthogonal polynomials. Despite the predicted bounds demanding no a priori information about the exact solution, they are validated to be spectrally accurate upon comparison with the explicit computational error accruing from the numerical solution of a variety of test problems, some chosen to be challenging to approximation methods, with known solutions. Potential limitations of the theory are discussed, but these are shown not to arise in the numerical computations.