Paper: Moving Least Square Reproducing Kernel Method Part II: Fourier Analysis
S. Li and W. K. Liu
"Computer Methods in Applied Mechanics and Engineering", Vol. 139, pp.
159-194, 1996
Abstract
In Part I of this work, the moving least square reproducing kernel (MLSRK)
method is reformulated and implemented. Based on its generic construction,
a m-consistency structure is discovered and the convergence theorems are
established. In this part of the work, a systematic Fourier analysis is
employed to evaluate and further establish the method. The preliminary
Fourier analysis reveals that MLSRK method is stable for sufficiently dense,
non-degenerated particle distribution, in the sense that the kernel function
family satisfies the Riesz bound. One of the novelties of the current approach
is to treat the MLSRK method as a variant of the ``standard'' finite element
method and depart from there to make a connection with the multiresolution
approximation. The highlight of this paper is to embrace the MLSRK formulation
with the notion of the controlled L_p-approximation. Based on its characterization,
the Strang-Fix condition for example, a systematic procedure is proposed
to design new window functions so they can enhance the computational performance
of the MLSRK algorithm. The main effort here is to obtain a constant correction
function in the interior region of a general domain. This can create a
leap in the approximation order of the MLSRK algorithm significantly, if
a highly smooth window function is embedded within the kernel. One consequence
of this development is the synchronized convergence phenomenon---a unique
convergence mechanism for the MLSRK method, i.e. by properly tuning the
dilation parameter, the convergence rate of higher order error norms will
approach to the same order convergence rate of the L_2 error norm---they
are synchronized.
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Professor Wing Kam Liu
Send email: w-liu@nwu.edu