Operators of Order Minus One

Department of Mathematics

Cornell University

Ithaca, NY 14853

Statistics Research Section

Australian National University

Canberra, ACT 2601

Australia

Applied Math Department

Brookhaven National Laboratory

Upton, NY 11973

Multigrid algorithms are developed to solve the discrete systems approximating
the solutions of operator equations involving pseudo-differential operators of
order minus one. Classical multigrid theory deals with the case of
differential operators of positive order. The pseudo-differential operator
gives rise to a coercive form on H^{\mhalf}(\d). Effective multigrid
algorithms are developed for this problem. These algorithms are novel in that
they use the inner product on H^{-1}(\d) as a base inner product for
the multigrid development. We show that the resulting rate of iterative
convergence can, at worst, depend linearly on the number of levels in these
novel multigrid algorithms. In addition, it is shown that the convergence
rate is independent of the number of levels (and unknowns) in the case of a
pseudo-differential operator defined by a single layer potential. Finally,
the results of numerical experiments illustrating the theory are presented.

Contributed April 2, 1993.