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> Volume 2, Issue 4
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Reinhold Schneider, Fb Mathematik THD, Darmstadt Guest

Posted: Fri Nov 29, 2002 2:27 pm Subject: Preprints available




Preprints available
Preprints available:
"Wavelet Approximation Methods for Pseudodifferential Equations I:
Stability and Convergence"
authors: Dahmen, W., RWTH Aachen
Proessdorf, S., IAAS Berlin
Schneider, R., TH Darmstadt,
Abstract:
This is the first part of two papers which are concerned with generalized
PetrovGalerkin schemes for elliptic periodic pseudodifferential equations
in R^n covering classical Galerkin methods, collocation and quasiinterpolation
These methods are based on a general setting of multiresolution analysis.
In this part we develop a general stability and convergence theory for such
a framework which recovers and extends many previously studied cases. The
key to the analysis is a local principle due to the second author. Its
applicability relies here on a sufficiently general version of a so called
discrete commutator property. These results establish important prerequisites
for developing and analysing in the second part methods for thefast solution
of the resulting linear systems. These methods are based on compressing the
stiffness matrices relative to wavelet bases for the given multiresolution
analysis.
"Wavelet Approximation Methods for Pseudodifferential Equations II:
Matrix Compression and Fast solution"
authors: Dahmen, W. , RWTH Aachen
Proessdorf, S. , IAAS Berlin
Schneider, R. , TH Darmstadt
Abstract:
This is the second part of two papers which are concerned with genralized
PetrovGalerkin schemes for elliptic periodic pseudodifferential equations
in R^n. This setting covers classical Galerkin methods, collocation and
quasiinterpolation. The numerical methods are based on a general framework
of multiresolution analysis, i.e., of sequences of nested spaces which are
generated by refinable functions. In this part we analyse compression
techniques for the resulting stiffness matrices relative to wavelet type
bases. We will show that, although the usual stiffness matrices are
generally not sparse, the order of the overall computational work which is
needed to realize a certain order of accuracy is of the form
${cal O} (N (log N )^b )$, where $N$ is the number of unknowns and
$b geq 0 $ is some real number (e.g. for fixed acuracy $b=0$).
Preprints are avaiable from: (after begining of April 1993)
Dr. Reinhold Schneider
Fb Mathematik THD
Schlossgartenstr.7
DW 6100 Darmstadt
Germany
email: schneider@mathematik.thdarmstadt.de 





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