
Classification of BSplinesWith the aid of the information, obtained in the grid generation process, the tensor product Bsplines are classified. Only those Bsplines, which have some support in the simulation domain, are relevant. The Figure below shows the relevant Bsplines 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 Bsplines are partitioned into outer and inner Bsplines, depending on the size of their support in the simulation region. As is illustrated in the Figure, some Bsplines 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 Bsplines are adjoined to the inner Bsplines 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 Bsplines form a stable basis with all the familiar properties of standard finite elements. The stabilization is necessary only locally, near the boundary. Bsplines, well in the interior of the domain, are not modified. In particular, for small grid width, the majority of inner Bsplines remains unchanged. Author: Joachim Wipper ; Last modification: 2006/08/31 09:38:56 UTC.
