Hierarchical conforming finite element methods for the biharmonic equation
SIAM Journal on Numerical Analysis
Multilevel algorithms considered as iterative methods on semidefinite systems
SIAM Journal on Scientific Computing
Journal of Computational and Applied Mathematics
Proceedings of the 27th annual conference on Computer graphics and interactive techniques
Piecewise Quadratic Approximations on Triangles
ACM Transactions on Mathematical Software (TOMS)
Wavelet Least Squares Methods for Boundary Value Problems
SIAM Journal on Numerical Analysis
Automatic construction of control triangles for subdivided Powell-Sabin splines
Computer Aided Geometric Design
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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 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 12-split element BPX preconditioner, although both methods have the same asymptotical complexity. Numerical experiments confirm this statement.