## Generalized multiple scale reproducing kernel particle methods.(English)Zbl 0896.76069

An approach to unify reproducing kernel methods and an extension to include time and spatial shifting are proposed. The groundwork is set by revisiting the Fourier analysis of discrete systems. The multiresolution concept, its significance in devising the reproducing kernel methods and its discrete counterpart, reproducing kernel particle methods, are explained. An edge detection technique based on multiresolution analysis is developed. This wavelet approach, together with particle methods, gives rise to a straightforward $$h$$-adaptivity algorithm. By using this framework, a Hermite reproducing kernel method is also proposed, and its relation to wavelet methods is presented. It is also shown that the new approach generalizes existing kernel methods, and it can easily be degenerated into other widely used methods. Finally, multiple-scale methods based on frequency and wave number shifting techniques are presented, together with a stability analysis for Newmark time-integration schemes for the low frequency equation.

### MSC:

 76M25 Other numerical methods (fluid mechanics) (MSC2010) 74S30 Other numerical methods in solid mechanics (MSC2010)
Full Text: