Wavelet and multiple scale reproducing kernel methods

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. This flexible space‐scale window function can be constructed to resemble the well‐known unstructured multigrid and hp ‐adaptive finite element methods. However, the multiple scale adaptive refinements are performed simply by inserting nodes into the highest wavelet scale solution region and at the same time narrowing the window function. Hence hp ‐like adaptive refinements can be performed without a mesh. An energy error ratio parameter is also introduced as a measure of aliasing error, and critical dilation parameters are determined for a class of spline window functions to obtain optimal accuracy. This optimal dilation parameter dictates the number of nodes covered under the support of a given window function. Numerical examples, which include the Helmholtz equation and the 1D and 2D 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.