The analysis of multigrid algorithms for pseudodifferential operators of order minus one

Abstract

Multigrid algorithms are developed to solve the discrete systems approximating the solutions of operator equations involving pseudodifferential operators of order minus one. Classical multigrid theory deals with the case of differential operators of positive order. The pseudodifferential operator gives rise to a coercive form on ${H^{ - 1/2}}(\Omega )$. Effective multigrid algorithms are developed for this problem. These algorithms are novel in that they use the inner product on ${H^{ - 1}}(\Omega )$ 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 pseudodifferential operator defined by a single-layer potential. Finally, the results of numerical experiments illustrating the theory are presented.