Wavelet approximation methods for pseudodifferential equations I: stability and convergence.
- Dahmen, Wolfgang
- Prößdorf, Sigfried
- Schneider, R.
2010 Mathematics Subject Classification
- 65R20 65N35 65N30 45E05 45E10 41A25 41A63 47G30
- Refinable functions, wavelets, periodic pseudodifferential operators, generalized Galerkin-Petrov schemes, discrete commutator property, stability analysis, convergence estimates
This is the first part of two papers which are concerned with generalized Petrov-Galerkin schemes for elliptic periodic pseudodifferential equations in ℝn covering classical Galerkin methods, collocation, and quasiinterpolation. These methods are based on a general setting of multiresolution analysis, i.e., of sequences of nested spaces which are generated by refinable functions. In this part we develop a general stability and convergence theory for such a framework which recovers and extends many previously studied special 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 mehods for the fast solution of the resulting linear systems. These methods are based on compressing the stiffness matrices relative to wavelet bases for the given multiresolution analysis.