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

发布时间：2018-08-21 12:13

【摘要】：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。
[Abstract]:In order to study the characteristics of spectral element method based on Least-Squares variation and Galerkin variation, the corresponding weak solutions for solving Poisson equation in annular region by using two variational methods in polar coordinate system are derived. The interpolation basis function is constructed by Chebyshev polynomial and the algebraic equations corresponding to two spectral element methods are obtained. The characteristics of the structure of the coefficient matrix are analyzed. The numerical results show that the new solution variable is introduced to reduce the order of the equation by the method of the Least-Squares spectral element method, which makes the form of algebraic equations more complicated, but the boundary condition is simpler than the Galerkin spectral element method. The two spectral element methods can solve the Poisson equation in polar coordinate system and obtain high accuracy numerical solution. The absolute error distribution of the two methods is basically the same, and the numerical error can be reduced by increasing the number of elements when the order of interpolation is fixed. And it shows the characteristics of algebraic precision, the speed of error decreasing is slow, and when the number of elements is fixed, the numerical error decreases faster with the increase of interpolation order in a certain range, which shows the characteristic of spectral precision, and when the interpolation order is higher in the unit, the numerical error decreases more quickly with the increase of interpolation order in a certain range. The condition number of the coefficient matrix of algebraic equations increases sharply, the system of equations is ill-conditioned and the numerical error increases, which limits the value of interpolation order in the unit. The research is valuable for further understanding the characteristics of the two spectral element methods in polar coordinate system for solving the Poisson equation and further using the correlation splitting algorithm to solve the actual flow problem.
【作者单位】：西安交通大学能源与动力工程学院;
【基金】：国家重点基础研究发展计划资助项目(2012CB026004)
【分类号】：O241.82

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