ERROR AND COMPLEXITY ANALYSIS FOR A COLLOCATION-GRID-PROJECTION PLUS
     PRECORRECTED-FFT ALGORITHM FOR SOLVING POTENTIAL INTEGRAL EQUATIONS
                      WITH LAPLACE OR HELMHOLTZ KERNELS

                                J. R. PHILLIPS

                                  Abstract.

In this paper we derive error bounds for a collocation-grid-projection scheme
tuned for use in multilevel methods for solving boundary-element
discretizations of potential integral equations.  The grid-projection scheme
is then combined with a precorrected-FFT style multilevel method for solving
potential integral equations with 1/r and e^{ik r}=r kernels.  A complexity
analysis of this combined method is given to show that for homogenous
problems, the method is order nlogn nearly independent of the kernel.  In
addition, it is shown analytically and experimentally that for an inhomogenity
generated by a very finely discretized surface, the combined method slows to
order n^{4/3}.  Finally, examples are given to show that the collocation-based
grid-projection plus precorrected-FFT scheme is competitive with
fast-multipole algorithms when considering realistic problems and 1/r kernels,
but can be used over a range of spatial frequencies with only a small
performance penalty.