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Wavelet approximation methods for pseudodifferential equations. I: Stability and convergence. (English) Zbl 0794.65082

This is the first of two papers which are concerned with generalized Petrov-Galerkin schemes for elliptic periodic pseudodifferential equations in 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.

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.

65N30Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods (BVP of PDE)
65N12Stability and convergence of numerical methods (BVP of PDE)
65N35Spectral, collocation and related methods (BVP of PDE)
42C40Wavelets and other special systems
35S15Boundary value problems for pseudodifferential operators
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