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Zbl 0813.65106
Amaratunga, Kevin; Williams, John R.; Qian, Sam; Weiss, John
Wavelet-Galerkin solutions for one-dimensional partial differential equations. (English)
[J] Int. J. Numer. Methods Eng. 37, No.16, 2703-2716 (1994). ISSN 0029-5981

This paper describes how wavelets can be used for solving partial differentiation equations by considering the one-dimensional counterpart of Helmholtz's equation. This technique necessitates the solution of linear systems of equations in the wavelet space rather than the physical space which implies a transform of the right-hand side into wavelet space and a transform of the solution back into physical space.\par Because, for this problem, the ensuing linear system is circulant it can be efficiently solved by a convolution approach and fast Fourier transforms. Numerical results suggest that wavelet solutions converge much faster than finite difference solutions and the gains in accuracy outweights the additional computation effort. In addition, because wavelets are localized in space, adaptive mesh refinement strategies can be efficiently implemented.
[K.Burrage (Brisbane)]

MSC 2000:: *65L10 Boundary value problems for ODE (numerical methods)
65L60 Finite numerical methods for ODE
34B05 Linear boundary value problems of ODE

Keywords: convergence; wavelets; Helmholtz's equation; fast Fourier transforms; adaptive mesh refinement

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