Multilevel evaluation of integral transforms with asymptotically smooth kernels

Multilevel algorithms developed for the fast evaluation of integral transforms and the solution of the corresponding integral and integrodifferential equations rely on smoothness properties of the discrete kernel (matrix) and thereby on grid uniformity (see [A. Brandt and A.A. Lubrecht, J. Comput. Phys., 90 (1990), pp. 348--370], [C.H. Venner and A.A. Lubrecht, Multigrid Methods IV: Proc. 4th European Multigrid Conference, Amsterdam1993, P. Hemker and P. Wesseling, eds., Birkhäuser, Basel, 1994]). However, in actual applications, e.g., in contact mechanics, in many cases a substantial increase of efficiency can be obtained using nonuniform 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 continuum kernel only, independent of the grid configuration. This will facilitate the introduction of local refinements, wherever needed. Also, the evaluations will generally be faster; for a d-dimensional problem only O(sd + 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 a fourth-order discretization. For testing purposes and to compare with results presented in [A. Brandt and A.A. Lubrecht, J. Comput. Phys., 90 (1990), pp. 348--370], uniform grids covering the entire domain were considered first.