Department of Mathematics

Cornell University

Ithaca, NY 14853

Applied Math Department

Brookhaven National Laboratory

Upton, NY 11973

We describe and analyze certain V-cycle multigrid algorithms with forms
defined by numerical quadrature applied to the approximation of symmetric
second order elliptic boundary value problems. This approach can be used for
the efficient solution of finite element systems resulting from numerical
quadrature as well as systems arising from finite difference discretizations.
The results are based on a regularity free theory and hence apply to meshes
with local grid refinement as well as the quasi-uniform case. It is shown
that uniform (independent of the number of levels) convergence rates often
hold for appropriately defined V-cycle algorithms with as few as one smoothing
per grid. These results hold even on applications without full elliptic
regularity, e.g., a domain in R^{2} with a crack.

Contributed April 2, 1993.