From the summary: We develop the hierarchical basis for meshless methods. A reproducing kernel hierarchical partition of unity is proposed in the framework of continuous representation, together with its discretized counterpart. To form such hierarchical partition, we introduce a class of basic wavelet functions. Then, based upon the built-in consistency conditions, we derive differential consistency conditions for the hierarchical kernel functions. It serves as an indispensable instrument in establishing the interpolation error estimate, which is theoretically proven and numerically validated. The new hierarchical basis is an intrinsic pseudo-spectral basis, which remains as a partition of unity in a local region, because the discrete wavelet kernels form a ‘partition of nullity’.
In part II, the hierarchical reproducing kernels are used as a multiple scale basis to compute numerical solutions of Helmholtz equation, to solve a model equation of wave propagation, and to simulate shear band formation in elasto-viscoplastic materials.