A. Brandt et Ch. Venner, MULTILEVEL EVALUATION OF INTEGRAL-TRANSFORMS WITH ASYMPTOTICALLY SMOOTH KERNELS, SIAM journal on scientific computing, 19(2), 1998, pp. 468-492
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.