University of Georgia
Mathematics Department Colloquium
Oleg Davydov
The University of Giessen, Germany
March 23, 2000
Smooth Piecewise Polynomial
Multiresolution Analysis on Irregular Triangulations
Abstract:
Both hierarchical bases and wavelets have as a prerequisite
a multiresolution analysis on the underlying nested sequence of
triangulations. The spaces of
piecewise polynomials are a natural tool for the construction of the
multiresolution analysis. In particular, if the bases are only required
to be continuous, usual finite-element spaces can be used.
In contrast to this, smooth conforming finite elements, such as Argyris
element,
are not suitable due to the fact that the corresponding spaces of
piecewise polynomials are not nested.
The idea to make use of
the basis of the full space of $C^1$-splines $S^1_d(\triangle)$ instead is
due to Dahmen, Oswald \& Shi (1994). However, the Morgan-Scott basis
for $S^1_d(\triangle)$ used in their work appears to be instable
if the triangulations have so-called near-degenerate edges and
near-singular vertices. Therefore, the triangulation has to be refined
in such a way (e.g., uniformly) that these geometrical configurations do not
appear.
We discuss a recent construction (joint work with L.~L.~Schumaker)
of a stable local basis for $C^1$ and, more general, $C^r$-splines, $r\ge1$,
which can be used to build stable multiresolution
analyses on arbitrary nested sequences of triangulations, with the only
restriction that the smallest angle of the triangles is controlled.
In particular, this construction applies to adaptively refined
triangulations.