Powell–Sabin spline based multilevel preconditioners for the biharmonic equation

The Powell–Sabin (PS) piecewise quadratic C 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 √ 3-refinement scheme instead of the traditional dyadic scheme. Topologically √ 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 12split element BPX preconditioner, although both methods have the same asymptotical complexity. Numerical experiments confirm this statement.