Multigrid Algorithms for C0 Interior Penalty Methods

Abstract

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.