Reproducing kernel hierarchical partition of unity, Part I - formulation and theory

This work is concerned with developing the hierarchical basis for meshless methods. A reproducing kernel hierarchical partition of unity is proposed in the framework of continuous representation as well as its discretized counterpart. To form such hierarchical partition, a class of basic wavelet functions are introduced. Based upon the built-in consistency conditions, the di erential consistency conditions for the hierarchical kernel functions are derived. It serves as an indispensable instrument in establishing the interpolation error estimate, which is theoretically proven and numerically validated. For a special interpolant with di erent combinations of the hierarchical kernels, a synchronized convergence e ect may be observed. Being di erent from the conventional Legendre function based p-type hierarchical basis, the new hierarchical basis is an intrinsic pseudo-spectral basis, which can remain as a partition of unity in a local region, because the discrete wavelet kernels form a `partition of nullityʹ. These newly developed kernels can be used as the multi- scale basis to solve partial di erential equations in numerical computation as a p-type re nement.