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Zbl 0794.65082
Dahmen, Wolfgang; Prössdorf, Siegfried; Schneider, R.
Wavelet approximation methods for pseudodifferential equations. I: Stability and convergence. (English)
[J] Math. Z. 215, No.4, 583-620 (1994). ISSN 0025-5874; ISSN 1432-1823/e

This is the first of two papers which are concerned with generalized Petrov-Galerkin schemes for elliptic periodic pseudodifferential equations in $\bbfR\sp n$ covering classical Galerkin methods, collocation, and quasi-interpolation. 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 the validity of a sufficiently general version of a so called discrete commutator property. This property is established for the present general setting by proving certain super- convergence results for the projectors defining the numerical schemes.\par These results provide important prerequisites for developing and analysing in a second paper [Wavelet approximation methods. II: Matrix compression and fast solution, Adv. Comput. Math. (to appear)] methods for the fast solution of the resulting linear systems. These methods are based on compressing the stiffness matrices relative to wavelet bases associated with the underlying multiresolution analysis.
[W.Dahmen (Aachen)]

MSC 2000:: *65N30 Finite numerical methods (BVP of PDE)
65N12 Stability and convergence of numerical methods (BVP of PDE)
65N35 Collocation methods (BVP of PDE)
42C40 Wavelets
35S15 Boundary value problems for pseudodifferential operators

Keywords: generalized Petrov-Galerkin schemes; elliptic periodic pseudodifferential equations; Galerkin methods; collocation; multiresolution analysis; general stability and convergence theory; discrete commutator propeerty; wavelet bases

Cited in: Zbl 0960.35114 Zbl 0899.65063 Zbl 0815.65115 Zbl 0826.65093

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