The Analysis of Multigrid Algorithms forPseudo -

Multigrid algorithms are developed to solve the discrete systems approximating the solutions of operator equations involving pseudo-diierential operators of order minus one. Classical multigrid theory deals with the case of diierential operators of positive order. The pseudo-diierential operator gives rise to a coercive form on H ?1=2 ((). EEective multigrid algorithms are developed for this problem. These algorithms are novel in that they use the inner product on H ?1 (() 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-diierential operator deened by a single layer potential. Finally, the results of numerical experiments illustrating the theory are presented.