Synchronized reproducing kernel interpolant via multiple wavelet expansion

In this paper, a new partition of unity – the synchronized reproducing kernel (SRK) interpolant – is derived. It is a class of meshless shape functions that exhibit synchronized convergence phenomenon: the convergence rate of the interpolation error of the higher order derivatives of the shape function can be tuned to be that of the shape function itself. This newly designed synchronized reproducing kernel interpolant is constructed as an series expansion of a scaling function kernel and the associated wavelet functions. These wavelet functions are constructed in a reproducing procedure, simultaneously with the scaling function kernel, by directly enforcing certain orders of vanishing moment conditions. To the authors knowledge, this unique interpolant is the first of its kind to be constructed, and to be used in numerical computations, both in concept and in practice. The new interpolants are in fact a group of special hierarchial meshless bases, and similar counterparts may exist in spline interpolation method, other meshless methods, Galerkin-wavelet method, as well as the finite element method. A detailed account of the subject is presented, and the mathematical principle behind the construction procedure is further elaborated. Another important discovery of this study is that the 1st order wavelet together with the scaling function kernel can be used as a weighting function in Petrov-Galerkin procedures to provide a stable numerical computation in some pathological problems. Benchmark problems in advection-diffusion problems, and Stokes flow problem are solved by using the synchronized reproducing kernel interpolant as the weighting function. Reasonably good results have been obtained. This may open the door for designing well behaved Galerkin procedures for numerical computations in various constrained media.