Paper: Moving Least Square Reproducing Kernel Method Part I: Methodology
and Convergence
W. K. Liu, S. Li and T. Belytschko
Accepted for publication in "Computer Methods in Applied Mechanics and
Engineering"
Abstract
This paper reformulates the moving least square interpolation scheme, in
a framework of the so-called moving least square kernel (MLSK) representation.
In this study, the procedure of constructing moving least square interpolation
function is facilitated by using the notion of reproducing kernel formulation,
which is a generalization of the early discrete approach. Here, the concept
of reproducing kernel integral representation is particularly emphasized,
such that the method not only shares its root with finite element method,
but also is lined with the multiresolution analysis in general. The ``shape
functions'' constructed by this method form a complete basis of finite
dimensional Hilbert space; moreover, the reproducing formula is not only
able to reproduce any m-th order polynomial exactly, but also able to reproduce
any smooth function accurately in a global least square sense. By tuning
the dilation parameter to a proper length, the subspace method that is
equipped with this particular interpolant function can achieve an optimal
accuracy, in a fixed particle distribution, to the response corresponding
to the source of a certain wave number or frequency band. An interpolation
error estimate is given to assess the convergence rate of the numerical
computation. It is shown that for sufficiently smooth function the interpolant
expansion in terms of sampled values will converge to the original function
in Sobolev norm. As a meshless method, the convergence rate is measured
by a new control variable --- dilation parameter ha of window function,
instead of the mesh size h as usually done in the finite element analysis.
To illustrate the technical procedure, convergence has been shown for the
numerical solution of second order elliptic differential equations in a
Galerkin procedure invoked with this interpolant. As an numerical example,
a two point boundary problem is solved by using the method, and a very
fast convergence rate is observed with respect to various norms.
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Professor Wing Kam Liu
Send email: w-liu@nwu.edu