Multigrid Methods for Hellan-Herrmann-Johnson Mixed Method of Kirchhoff Plate Bending Problems

A V-cycle multigrid method for the Hellan-Herrmann-Johnson (HHJ) discretization of the Kirchhoff plate bending problems is developed in this paper. It is shown that the contraction number of the V-cycle multigrid HHJ method is bounded away from one uniformly with respect to the mesh-size. The key is a stable decomposition of the kernel space which is derived from an exact sequence of the HHJ method. The uniform convergence is achieved for V-cycle multigrid method with only one smoothing step and without full elliptic regularity. Some numerical experiments are provided to confirm the proposed V-cycle multigrid method. The exact sequences of the HHJ method and the corresponding commutative diagram is of some interest independent of the current context.