Wavelet-Galerkin solutions for one-dimensional partial differential equations.

*(English)*Zbl 0813.65106This 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.

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.

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.

Reviewer: K.Burrage (Brisbane)

##### MSC:

65L10 | Numerical solution of boundary value problems involving ordinary differential equations |

65L60 | Finite element, Rayleigh-Ritz, Galerkin and collocation methods for ordinary differential equations |

34B05 | Linear boundary value problems for ordinary differential equations |

##### Keywords:

convergence; wavelets; Helmholtz’s equation; fast Fourier transforms; adaptive mesh refinement
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\textit{K. Amaratunga} et al., Int. J. Numer. Methods Eng. 37, No. 16, 2703--2716 (1994; Zbl 0813.65106)

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