基于Coiflet的二维小波有限元构造与应用

Construction and application of two-dimentional wavelet finite element based on Coiflet

使用Coiflet小波尺度函数作为插值函数,构造了一种基于零矩尺度函数的小波有限元,同时构造了二维Coiflet小波,并以其尺度函数作为插值函数构造了二维小波有限元.由于Coiflet小波同时具有尺度函数消失矩和小波函数消失矩,因此简化了移动矩计算方法,使移动矩和联系系数等相关计算更方便、准确.引入了转换矩阵,实现从小波空间到物理空间的转换.以一个一维微分方程和二维赫尔姆兹方程为例,使用本文方法对这两个问题进行了数值求解,结果表明Coiflet有限元法能够得到很高的精度.

In this work, Coiflet wavelet element was constructed using Coiflet scaling functions as interpolating function. At the same time, two-dimensional wavelet finite element and construct two-dimensional wavelet finite element were constructed using its scaling functions. The algorithm of translated moments was predigested to make relative calculations, such as translated moments, connection coefficient and so on, and which are more convenient and precise, because Coiflet wavelet has the vanishing moments of both the wavelet function and the scaling function. A transformation matrix was introduced for the transformation between wavelet spaces and physical spaces. Two numerical examples were put forward, a one-dimensional differential equation and a two-dimentional Helmholtz equation, to solve the equations using the method give in this work. The results show that Coiflet wavelet finite element has high accuracy.