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Wavelet and multiscale methods for operator equations. (English) Zbl 0884.65106

Iserles, A. (ed.), Acta Numerica Vol. 6, 1997. Cambridge: Cambridge University Press. 55-228 (1997).
This paper gives a very clear overview of recent developments in the use of wavelet methods in the numerical solution of integral equations and partial differential equations. It also points out some of the remaining difficulties, particularly, the question of how to construct wavelets on domains of fairly general shape. It isn’t even clear what wavelets one ought to use on a ball or a sphere. Also, much work remains to be done in the areas of wavelet-based methods for nonlinear partial differential equations and for singularly-perturbed convection-diffusion equations. Most of the paper, however, is devoted to the significant strides that have been made in the applications to more regular elliptic and parabolic problems.
For the entire collection see [Zbl 0868.00024].

MSC:

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35J65 Nonlinear boundary value problems for linear elliptic equations
35K55 Nonlinear parabolic equations
45G10 Other nonlinear integral equations
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
65R20 Numerical methods for integral equations
42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
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