Reproducing kernel hierarchical partition of unity, part II - Applications

In this part of the work, the meshless hierarchical partition of unity proposed in [1], referred here as Part I, is used as a multiple scale basis in numerical computations to solve practical problems. The applications discussed in the present work fall into two categories: (1) a wavelet adaptivity refinement procedure; and (2) a so-called wavelet Petrov-Galerkin procedure. In the applications of wavelet adaptivity, the hierarchical reproducing kernels are used as a multiple scale basis to compute the numerical solutions of the Heimholtz equation, a model equation of wave propagation problems, and to simulate shear band formation in an elasto-viscoplastic material, a problem dictated by the presence of the high gradient deformation. In both numerical experiments, numerical solutions with high resolution are obtained by inserting the wavelet-like basis into the primary interpolation function basis, a process that may be viewed as a spectral p-type refinement. By using the interpolant that has synchronized convergence property as a weighting function, a wavelet Petrov-Galerkin procedure is proposed to stabilize computations of some pathological problems in numerical computations, such as advection-diffusion problems and Stokes' flow problem; it offers an alternative procedure in stabilized methods and also provides some insight, or new interpretation of the method. Detailed analysis has been carried out on the stability and convergence of the wavelet Petrov-Galerkin method.