Classification of B-Splines

With the aid of the information, obtained in the grid generation process, the tensor product B-splines are classified.

Only those B-splines, which have some support in the simulation domain, are relevant. The Figure below shows the relevant B-splines of order 3 for a planar domain. They are marked with dots at the lower left corner of their support; the boundary of a support is shown at the bottom left.

The relevant B-splines are partitioned into outer and inner B-splines, depending on the size of their support in the simulation region.

As is illustrated in the Figure, some B-splines can have only very little relevant support. They cause instabilities in the numerical solution; but, cannot be omitted without affecting the approximation order.

Therefore, the outer B-splines are adjoined to the inner B-splines by forming appropriate linear combinations. For the quality of the finite element approximation the correct choice of the coupling coefficients is essential. Local support and full approximation order must be retained. The resulting extended B-splines form a stable basis with all the familiar properties of standard finite elements.

The stabilization is necessary only locally, near the boundary. B-splines, well in the interior of the domain, are not modified. In particular, for small grid width, the majority of inner B-splines remains unchanged.

Author: Joachim Wipper ; Last modification: 2006/08/31 09:38:56 UTC.