Moving Least Square Reproducing Kernel Method ( III ) : Wavelet Packet & Its

This work is a natural extension of the work done in Part II of this series. A new partition of unity | the synchronized reproducing kernel (SRK) interpolant|is proposed within the framework of moving least square reproducing kernel representation. It is a further development and generalization of the reproducing kernel particle method (RKPM), which demonstrates some superior computational capability in multiple scale numerical simulations. To form such an interpolant, a class of new wavelet functions are introduced in an unconventional way, and they form an independent sequence that is referred to as the wavelet packet. By choosing different combinations in the wavelet series expansion, the desirable synchronized convergence eeect in interpolation can be achieved. Based upon the built-in consistency conditions, the diierential consistency conditions for the wavelet functions are derived. It serves as an indispensable instrument in establishing the interpolation error estimate, which is theoretically proven and numerically validated. The procedure of generating wavelet functions in this study is remarkably simple, eecient, and quite diierent from the usual wavelet analysis. Several examples of such wavelets are constructed to illustrate their intrinsic properties. A wavelet Petrov-Galerkin procedure is also proposed using the wavelet packet to stabilize the numerical computation. Detailed analysis has been carried out on the stability and convergence of the numerical solution of the advective-diiusive equation obtained by wavelet Petrov-Galerkin method. Finally, a wavelet partition of unity, which can be viewed as a special case of p-reenement, is also formulated in the context of meshless method as well as the nite element method.