# zbMATH — the first resource for mathematics

Filters, reproducing kernel, and adaptive meshfree method. (English) Zbl 1038.74681
Summary: Reproducing kernel, with its intrinsic feature of moving averaging, can be utilized as a low-pass filter with scale decomposition capability. The discrete convolution of two $$n$$th order reproducing kernels with arbitrary support size in each kernel results in a filtered reproducing kernel function that has the same reproducing order. This property is utilized to separate the numerical solution into an unfiltered lower order portion and a filtered higher order portion. As such, the corresponding high-pass filter of this reproducing kernel filter can be used to identify the locations of high gradient, and consequently serves as an operator for error indication in meshfree analysis. In conjunction with the naturally conforming property of the reproducing kernel approximation, a meshfree adaptivity method is also proposed.

##### MSC:
 74S30 Other numerical methods in solid mechanics (MSC2010) 74K20 Plates