中文摘要：

具有切向边界的无散度小波在向量场的数值模拟中扮演着重要的角色.鉴于Hardin-Marasovich小波函数的零边值性质和简单结构,主要研究一类利用Hardin-Marasovich小波函数构造的具有切向边界的三维各向同性无散度多小波。首先,基于Hardin-Marasovich小波函数的微分关系,证明了具有切向边界的无散度向量场在对应的向量尺度空间上的双正交投影还是无散度的。其次,利用无散度空间的刻画给出了各向同性无散度尺度函数的定义,并证明对应的无散度尺度函数空间构成了一个无散度多尺度分析。最后,定义各向同性无散度多小波,给出切向边界无散度向量在无散度小波基下分解系数与经典小波基下分解系数的关系,从而说明无散度向量的小波分解系数可快速计算。

英文摘要：

Divergence-free wavelets with tangential boundary plays an important role in numerical simulation of vector fields.In view of the zero boundary and the simple structure of Hardin-Marasovich wavelets,a class of three-dimensional isotropic divergence-free wavelets with tangential boundary are studied.Firstly,based on differential relations of Hardin-Marasovich wavelet functions,it is proved that the bi-orthogonal projection of divergence-free vector fields is still divergence-free.Then,the definition of isotropic divergence-free scale functions are given based on the characterization of divergence-free space,and the corresponding divergence-free scale spaces are proved to form a multiresolution analysis.Finally,the isotropic divergence-free multiwavelets are defined,and the relation between the decomposition coefficients of the divergence-free wavelets and the classical wavelets is given,which shows that the divergence-free decomposition coefficients can be fastly computed.