Analysis of V - Cycle Multigrid

We describe and analyze certain V-cycle multigrid algorithms with forms deened by numerical quadrature applied to the approximation of symmetric second order elliptic boundary value problems. This approach can be used for the eecient solution of nite element systems resulting from numerical quadrature as well as systems arising from nite diierence discretizations. The results are based on a regularity free theory and hence apply to meshes with local grid reenement as well as the quasi-uniform case. It is shown that uniform (independent of the number of levels) convergence rates often hold for appropriately deened 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.