Multilevel Evaluation of Integral Transforms on Adaptive Grids

Fast numerical evaluation of integral transforms on an adaptive grid, i.e. using local grid refinement, requires an algorithm that relies on smoothness properties only of the continuum kernel, independent of its discrete form. The basic outline of such an algorithm was given in [6], where it was shown that already on a uniform grid this algorithm was more efficient than earlier fast evaluation algorithms [4, 5]. In this paper we outline its detailed formulation for the actual case of local grid refinements. Numerical results are presented for a model problem with a singularity. First it is shown that on a uniform grid this singularity dictates a much deteriorated relation between work and accuracy in comparison with the regular case (where accuracy is measured in terms of approximating the continuum transform, of course). Next we demonstrate that with the presented fast evaluation algorithm on a non-uniform grid one can restore the regular work to accuracy relation, i.e., obtain the same efficiency as for the case without a singularity.