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Moving least-square reproducing kernel methods. I: Methodology and convergence. (English) Zbl 0883.65088
Summary: 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 $$m$$th 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 a 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 $$\varrho$$ 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.

##### MSC:
 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs 35J25 Boundary value problems for second-order elliptic equations 65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs 65N15 Error bounds for boundary value problems involving PDEs
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