基于Hermite插值的解耦小波基的构造及应用

Construction and application of the decoupling wavelet bases based on Hermite interpolation

小波的多分辨分析特性使得我们可以采用逐级嵌套,不断扩大的一系列空间来逼近一个函数,建立一个多分辨的有限元空间.但是,在多分辨的有限元空间中的不同尺度空间之间常常存在耦合项,如何消除这些耦合项以构造更为有效的自适应算法仍然是一个需要深入研究的问题.对于欧拉梁问题,利用小波的消失矩与多项式的正交关系,构造了基于Hermite插值的解耦小波基及相应的小波单元,消除了低分辨逼近空间与细节空间以及各个细节空间之间耦合项,实现了基于空间尺度解耦的自适应算法.数值算例验证了该方法的有效性.

Using the multi-resolution analysis characteristics of wavelet, a function may be ex- pressed by a sequence of spaces which are nested and constantly expanded, and a multi-resolution finite element space may be established. However, there are always couplings across scales in the multi-resolution finite element space. How to eliminate the couplings is still a problem which nee- ded to be studied further. For the Euler beam, the decoupling wavelet bases and the correspond- ing wavelet element based on Hermite interpolation were developed to solve Euler beam problem by using the orthogonality between the vanishing moment of wavelet and polynomial. The decou- pling wavelet bases may eliminate the couplings between the low resolution space and detail spaces and that among the detail spaces in different scales. Then an adaptive algorithm based on scale-decoupling was constructed. The numerical example verifies the effectiveness of the pro- posed method.