Convergence of nonconforming multigrid methods without full elliptic regularity

We consider nonconforming multigrid methods for symmetric positive definite second and fourth order elliptic boundary value problems which do not have full elliptic regularity. We prove that there is a bound (< 1) for the contraction number of the W-cycle algorithm which is independent of mesh level, provided that the number of smoothing steps is sufficiently large. We also show that the symmetric variable V-cycle algorithm is an optimal preconditioner.