Multiscale methods and wavelets – concepts and applications. (Multiskalen-Methoden and Wavelets – Konzepte und Anwendungen.) (German) Zbl 0835.65126

The paper gives an overview over applications of multiscale methods in numerical mathematics and sketches further developments in this direction. Especially, it addresses the preconditioning of stiffness matrices arising in the Galerkin solution of elliptic partial differential equations and the compression (preconditioning) of full- populated matrices appearing in the Petrov-Galerkin solution of special integral- and pseudo-differential equations.
For details, the author suggests reference to other papers on the same topic, in many cases written by himself and coauthors.


65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65J10 Numerical solutions to equations with linear operators
65-02 Research exposition (monographs, survey articles) pertaining to numerical analysis
65F10 Iterative numerical methods for linear systems
65R20 Numerical methods for integral equations
65F35 Numerical computation of matrix norms, conditioning, scaling
35S15 Boundary value problems for PDEs with pseudodifferential operators
35J25 Boundary value problems for second-order elliptic equations
42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
41A17 Inequalities in approximation (Bernstein, Jackson, Nikol’skiń≠-type inequalities)
47G30 Pseudodifferential operators
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
47A50 Equations and inequalities involving linear operators, with vector unknowns