×

zbMATH — the first resource for mathematics

Wavelet and multiple scale reproducing kernel methods. (English) Zbl 0885.76078
Multiple scale methods based on reproducing kernel and wavelet analysis are developed. These permit the response of a system to be separated into different scales. These scales can be either the wave numbers corresponding to spatial variables or the frequencies corresponding to temporal variables, and each scale response can be examined separately. This complete characterization of the unknown response is performed through the integral window transform, and a space-scale and time-frequency localization process is achieved by dilating the flexible multiple scale window function. An error estimation technique based on this decomposition algorithm is developed which is especially useful for local mesh refinement and convergence studies. Numerical examples, which include the Helmholtz equation and the one- and two-dimensional advection-diffusion equations, are presented to illustrate the high accuracy of the methods using the optimal dilation parameter, the concept of multiresolution analysis, and the meshless unstructured adaptive refinements.

MSC:
76M25 Other numerical methods (fluid mechanics) (MSC2010)
76R99 Diffusion and convection
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Liu, Int. j. numer. methods fluids 20 pp 1081– (1995)
[2] and , ’Reproducing kernel particle methods for elastic and plastic problems’, in and (eds), Advanced Computational Methods for Material Modeling, AMD, Vol. 180 and PVP Vol. 268, ASME, New York, 1993, pp. 175-190.
[3] Liu, Int. J. Numer. Methods Eng. 38 pp 1655– (1995)
[4] and , ’Reproducing kernel and wavelet particle methods’, in and (eds), Aerospace Structures: Nonlinear Dynamics and System Response, AD Vol. 33, ASME, New York, 1993, pp. 39-56.
[5] An Introduction to Wavelets, Academic, New York, 1992. · Zbl 0925.42016
[6] Daubechies, CBMS/NSF Series in Applied Mathematics, No. 61, Ten Lectures on Wavelets, SIAM, Philadelphia, PA, 1992.
[7] and , Signals and Systems, 2nd edn, PWS-Kent, 1987.
[8] Hughes, Comput. Methods Appl. Mech. Eng. 73 pp 173– (1989)
[9] Shakib, Comput. Methods Appl. Mech. Eng. 87 pp 35– (1991)
[10] Wavelets: A Tutorial in Theory and Applications, Academic, New York, 1992. · Zbl 0744.00020
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.