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Multiwavelet approximation methods for pseudodifferential equations on curves. Stability and convergence analysis

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Abstract

We develop a stability and convergence analysis of Galerkin–Petrov schemes based on a general setting of multiresolution generated by several refinable functions for the numerical solution of pseudodifferential equations on smooth closed curves. Particular realizations of such a multiresolution analysis are trial spaces generated by biorthogonal wavelets or by splines with multiple knots. The main result presents necessary and sufficient conditions for the stability of the numerical method in terms of the principal symbol of the pseudodifferential operator and the Fourier transforms of the generating multiscaling functions as well as of the test functionals. Moreover, optimal convergence rates for the approximate solutions in a range of Sobolev spaces are established.

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  1. Weierstraß-Institut für Angewandte Analysis und Stochastik, Mohrenstrasse 39, D-10117, Berlin, Germany

    Siegfried Prössdorf & Jörg Schult

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  1. Siegfried Prössdorf
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  2. Jörg Schult
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Prössdorf, S., Schult, J. Multiwavelet approximation methods for pseudodifferential equations on curves. Stability and convergence analysis. Advances in Computational Mathematics 9, 145–171 (1998). https://doi.org/10.1023/A:1018977120831

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