Least-Squares及Galerkin谱元方法求解环形区域内的泊松方程

Least-Squares and Galerkin Spectral Element Methods for Poisson Equation in Annular Regions

为研究基于Least-Squares变分及Galerkin变分两种形式的谱元方法的求解特性,推导了极坐标系中采用两种变分方法求解环形区域内Poisson方程时对应的弱解形式,采用Chebyshev多项式构造插值基函数进行空间离散,得到两种谱元方法对应的代数方程组,由此分析了系数矩阵结构的特点。数值计算结果显示:Least-Squares谱元方法为实现方程的降阶而引入新的求解变量,使得代数方程组形式更为复杂,但边界条件的处理比Galerkin谱元方法更为简单;两种谱元方法均能求解极坐标系中的Poisson方程且能获得高精度的数值解,二者绝对误差分布基本一致;固定单元内的插值阶数时,增加单元数可减小数值误差,且表现出代数精度的特点,误差降低速度较慢,而固定单元数时,在一定范围内数值误差随插值阶数的增加而减小的速度更快,表现出谱精度的特点;单元内插值阶数较高时,代数方程组系数矩阵的条件数急剧增多,方程组呈现病态,数值误差增大,这一特点限制了单元内插值阶数的取值。研究内容对深入了解两种谱元方法在极坐标系中求解Poisson方程时的特点、进一步采用相关分裂算法求解实际流动问题具有参考价值。

To investigate the characteristics of spectral element methods based on Least-Squares and Galerkin variation, two different kinds of weak formulation for Poisson equation in the polar coordinate are presented. After discretization by the Chebyshev basis function, the corresponding algebraic equations are obtained, and the structures of the coefficient matrixes are analyzed. It is clear that the algebraic equations of Least-Squares spectral element method are more complex than Galerkin spectral element method due to the introduced auxiliary variables for modifying second-order partial differential equations into first-order system. However, the boundary conditions can be more easily dealt with via Least-Squares spectral element method. Numerical results show that both spectral element methods can get high-order numerical accuracy and the numerical errors keep almost consistent. When the interpolation order in each element is fixed, the numerical errors slowly decrease by refining the elements, and the algebraic accuracy can be obtained. A faster decay of the numerical errors can be observed by heightening the interpolation order to demonstrate the spectral accuracy property when the total elements are fixed. However, if the interpolation order gets a larger value, the numerical errors increase unexpectedly for the fast increasing condition number of the algebraic equations. This research may facilitate understanding these two different spectral element methods for the Poisson equation in polar coordinate, and further solving flow problems with splitting algorithm.