Wavelet and multiple scale reproducing kernel methods.

*(English)*Zbl 0885.76078Multiple 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 |

##### Keywords:

integral window transform; error estimation technique; decomposition algorithm; Helmholtz equation; advection-diffusion equations; optimal dilation parameter; multiresolution analysis; meshless unstructured adaptive refinements
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\textit{W. K. Liu} and \textit{Y. Chen}, Int. J. Numer. Methods Fluids 21, No. 10, 901--931 (1995; Zbl 0885.76078)

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