圆柱体上二阶椭圆特征值问题的有效Galerkin谱逼近

An effective spectral-Galerkin method for second order elliptic eigenvalue problems in cylindrical geometries

提出了三维柱体域上二阶椭圆特征值问题的一种新的谱逼近方法.首先,将直角坐标系下的二阶椭圆特征值问题转化为柱坐标系下的等价形式,再利用变量分离方法将原问题转化为矩形区域上的一系列二维特征值问题;其次,针对实心圆柱体和空心圆柱体两种情况,分别引入了两种Sobolev空间和相应的多项式逼近空间,对每个降维的二维特征值问题建立变分形式和离散格式;最后,利用全连续算子的谱理论和非一致带权Sobolev空间中投影算子的逼近性质证明了逼近解的误差估计.另外,通过构造逼近空间的一组有效基函数,推导了离散格式基于张量积的矩阵形式.数值结果表明我们的算法是有效的和高精度的.

In this paper,a new spectral approximation method for second order elliptic eigenvalue problem in three-dimensional cylindrical geometries is proposed.Firstly,the second order elliptic eigenvalue problem in rectangular coordinate system is transformed into an equivalent form in cylindrical coordinate system,and then the original problem is transformed into a series of two-dimensional eigenvalue problems in rectangular region by variable separation method.Secondly,for the two cases of solid cylinder and hollow cylinder,two Sobolev spaces and corresponding polynomial approximation spaces are introduced respectively,and the weak forms and discrete schemes are established for each reduced dimensional two-dimensional eigenvalue problems.The error estimation of the approximate solution is proved by using the spectrum theory of fully continuous operators and the approximation properties of projection operators in non-uniform weighted Sobolev spaces.In addition,a set of effective basis functions are constructed in the approximation spaces,and the matrix forms of the discrete schemes based on tensor product are also derived.Some numerical examples show that our algorithm is effective and high accurate.