Multiscale methods for pseudodifferential equations. (English) Zbl 0788.65112

Dahmen, Wolfgang; Prössdorf, Siegfried; Schneider, R.

Schumaker, Larry L. (ed.) et al., Recent advances in wavelet analysis. Boston, MA: Academic Press, Inc.. Wavelet Anal. Appl. 3, 191-235 (1994).

Without giving proofs the authors summarize their papers concerning the use of wavelets in the fast solution of linear pseudo-differential equations enclosing boundary integral equations as an important ingredient.
For the periodic case they consider numerical methods as Galerkin-Petrov or collocation methods in a unified way. Main topics are stability and error analysis in nested trial spaces of wavelets and preconditioning and matrix compression by changing to multiscale bases. Finally, the authors discuss the numerical effort of their approach.
For the entire collection see [Zbl 0782.00090].

Reviewer: G.Bruckner (Berlin)

MSC:

65N55	Multigrid methods; domain decomposition for boundary value problems involving PDEs
65N35	Spectral, collocation and related methods for boundary value problems involving PDEs
65N38	Boundary element methods for boundary value problems involving PDEs
65N30	Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N12	Stability and convergence of numerical methods for boundary value problems involving PDEs
65N15	Error bounds for boundary value problems involving PDEs
35S05	Pseudodifferential operators as generalizations of partial differential operators

Keywords:

Galerkin-Petrov method; wavelets; linear pseudo-differential equations; boundary integral equations; collocation methods; stability; error analysis; preconditioning