Powell-Sabin spline based multilevel preconditioners for the biharmonic equation

The Powell-Sabin (PS) piecewise quadratic C(1) finite element on the PS 12-split of a triangulation is a common choice for the construction of a BPX-type preconditioner for the biharmonic equation. In this note we investigate the related Powell-Sabin element on the PS 6-split instead of the PS 12-split for the construction of such preconditioners. For the PS 6-split element multilevel spaces can be created using a root 3-refinement scheme instead of the traditional dyadic scheme. Topologically root 3-refinement has many advantages: it is a slower refinement than the dyadic split operation, and it alternates the orientation of the refined triangles. Therefore we expect a reduction of the amount of work when compared to the PS 12-split element BPX preconditioner, although both methods have the same asymptotical complexity. Numerical experiments confirm this statement. (C) 2010 IMACS. Published by Elsevier B.V. All rights reserved.