A PhD thesis @ NALAG

TW 2008_03

Hendrik Speleers
Construction, analysis and application of Powell-Sabin spline finite elements

Advisor(s): Paul Dierckx, Stefan Vandewalle

Abstract

Powell-Sabin splines are piecewise quadratic polynomials with a global C1-continuity. They are defined on triangulations, and admit a compact representation in a normalized spline basis. These splines have been used successfully in the area of computer aided geometric design for the modelling and fitting of surfaces. In this thesis we examine the applicability of Powell-Sabin splines for the numerical solution of partial differential equations. Because of the inherent higher order of continuity, a finite element discretization based on Powell-Sabin splines is typically more efficient than a similar discretization with classical Lagrange finite elements.

Special emphasis goes to adaptive refinement. We elaborate a local subdivision algorithm for Powell-Sabin splines. The PS triangulation is locally refined with a sqrt(3)-scheme. To avoid the construction of poorly shaped triangles, a heuristic for refinement propagation is introduced. As alternative, we develop a hierarchical variant of Powell- Sabin splines, which allows local refinement in a natural way. A quasi-hierarchical basis is constructed that satisfies similar properties as the Powell-Sabin spline basis: the basis functions have a local support, they form a convex partition of unity, they are C1-continuous, they are stable, and they have a geometrically intuitive interpretation involving control triangles.

We present a multigrid algorithm for the efficient solution of the linear systems of equations that arise from the finite element discretization. Using calculations on a hierarchy of coarser meshes, the convergence of a basic iterative solver can be accelerated.

Finally, we investigate how to impose boundary conditions on Powell-Sabin splines. By choosing particular basis functions at the boundary, the set of constraints on the spline coefficients can be simplified.