Additive Multigrid Theory

Abstract

The convergence of the V-cycle multigrid algorithm is usually handled by a multiplicative theory where the iteration operator (matrix) is expressed as a product of operators measuring the effect of the multigrid algorithm on different grid levels.

In this talk an additive convergence theory for the V-cycle algorithm will be presented. This theory is effective for establishing the asymptotic behavior of the contraction number of the V-cycle algorithm as the number of smoothing steps is increased.

The following applications of the additive theory will be discussed:
(1) a complete generalization of the classical V-cycle convergence theorem of Braess and Hackbusch to the case of less than full elliptic regularity,
(2) convergence of V-cycle and F-cycle algorithms for nonconforming methods with a sufficiently large number of smoothing steps.