MULTILEVEL EVALUATION OF INTEGRAL-TRANSFORMS WITH ASYMPTOTICALLY SMOOTH KERNELS

Multilevel algorithms developed for the fast evaluation of integral tr ansforms and the solution of the corresponding integral and integrodif ferential equations rely on smoothness properties of the discrete kern el (matrix) and thereby on grid uniformity (see [A. Brandt and A.A. Lu brecht, J. Comput. Phys., 90 (1990), pp. 348-370], [C.H. Venner and A. A. Lubrecht, Multigrid Methods IV: Proc. 4th European Multigrid Confer ence, Amsterdam 1993, P. Hemker and P. Wesseling, eds., Birkhauser, Ba sel, 1994]). However, in actual applications, e.g., in contact mechani cs, in many cases a substantial increase of efficiency can be obtained using nonuniform grids, since the solution is smooth in large parts o f the domain with large gradients that occur only locally. In this pap er a new algorithm is presented which relies on the smoothness of the continuum kernel only, independent of the grid configuration. This wil l facilitate the introduction of local refinements, wherever needed. A lso, the evaluations will generally be faster; for a d-dimensional pro blem only O(s(d+1)) operations per gridpoint are needed if s is the or der of discretization. The algorithm is tested using a one-dimensional model problem with logarithmic kernel. Results are presented using bo th a second-and a fourth-order discretization. For testing purposes an d to compare with results presented in [A. Brandt and A.A. Lubrecht, J . Comput. Phys., 90 (1990), pp. 348-370], uniform grids covering the e ntire domain were considered first.