Multilevel Evaluation of Integral Transforms Withasymptotically Smooth

Multilevel algorithms developed for the fast evaluation of integral transforms and the solution of the corresponding integral and integro-diierentialequations rely on smoothnessproperties of the discrete kernel (matrix) and thereby on grid uniformity (see 6], 18]). However, in actual applications, e.g. in contact mechanics, in many cases a substantial increase of eeciency can be obtained using non-uniform grids, since the solution is smooth in large parts of the domain with large gradients that occur only locally. In this paper a new algorithm is presented which relies on the smoothness of the continuumkernel only, independent of the grid connguration. This will facilitate the introduction of local reenements, wherever needed. Also, the evaluations will generally be faster; for a d dimensional problem only O(s d+1) operations per gridpoint are needed, if s is the order of discretization. The algorithm is tested using a one dimensional model problem with logarithmic kernel. Results are presented using both a second and fourth order discretization. For testing purposes, and to compare with results presented in 6], uniform grids covering the entire domain were considered rst.