Multigrid algorithms for C0 interior penalty methods

Multigrid algorithms for C-0 interior penalty methods for fourth order elliptic boundary value problems on polygonal domains are studied in this paper. It is shown that V-cycle, F-cycle and W-cycle algorithms are contractions if the number of smoothing steps is sufficiently large. The contraction numbers of these algorithms are bounded by Cm-alpha, where m is the number of presmoothing (and postsmoothing) steps, alpha is the index of elliptic regularity, and the positive constant C is mesh-independent. These estimates are established for a smoothing scheme that uses a Poisson solve as a preconditioner, which can be easily implemented because the C-0 finite element spaces are standard spaces for second order problems. Furthermore the variable V-cycle algorithm is also shown to be an optimal preconditioner.