Moving least-square reproducing kernel methods (I) methodology and convergence

This paper formulates the moving least-square interpolation scheme in a framework of the so-called moving least-square reproducing kernel (MLSRK) representation. In this study, the procedure of constructing moving least square interpolation function is facilitated by using the notion of reproducing kernel formulation, which, as a generalization of the early discrete approach, establishes a continuous basis for a partition of unity. This new formulation possesses the quality of simplicity, and it is easy to implement. Moreover, the reproducing kernel formula proposed is not only able to reproduce any mth order polynomial exactly on an irregular particle distribution, but also serves as a projection operator that can approximate any smooth function globally with an optimal accuracy. In this contribution, a generic m-consistency relation has been found, which is the essential property of the MLSRK approximation. An interpolation error estimate is given to assess the convergence rate of the approximation. It is shown that for sufficiently smooth function the interpolant expansion in terms of sampled values will converge to the original function in the Sobolev norms. As a meshless method, the convergence rate is measured by a new control variable - dilation parameter ρ of the window function, instead of the mesh size h as usually done in the finite element analysis. To illustrate the procedure, convergence has been shown for the numerical solution of the second-order elliptic differential equations in a Galerkin procedure invoked with this interpolant. In the numerical example, a two point boundary problem is solved by using the method, and an optimal convergence rate is observed with respect to various norms.