极坐标系下的Legendre谱元方法求解Poisson-型方程

摘要

The key to solve Poisson-type equations in polar coordinates is its singularity at r=0.In this paper,a Legendre spectral element method (SEM) based on the Galerkin variational formulation for solving the Poisson-type equations in polar coordinates was proposed.The physical domain was divided into a number of elements and the Legendre polynomials were adopted in every computational element.Further,the Legendre-Gauss-Radau (LGR) quadrature points were used in the elements which involved the origin while Legendre-Gauss-Lobatto (LGL) quadrature points in the others in the radial direction,so that the 1/r coordinate singularity was avoided successfully.As to the azimuthal direction,the LGL quadrature points were employed.The clustering of collocation points near the pole could be prevented through the technique of domain decomposition.Finally,the method was applied to several Poisson-type equations subject to a Dirichlet or Neumann boundary condition.The numerical results demonstrate that the SEM has high accuracy.%r=0处的坐标奇异性是求解极坐标下Poisson-型方程的关键。本文提出一种极坐标系下基于Galerkin变分的Legendre谱元方法用于求解圆形区域内的Poisson-型方程,物理区域的径向和周向划分若干单元,计算单元均采用Legendre多项式展开;圆心所在单元的径向使用LGR（Legendre Gauss Radau）积分点,其他单元径向使用LGL（Legendre Gauss Lobatto）积分点,从而避免了极点处1/r坐标奇异性,周向单元均采用LGL积分点。利用区域分解技术,可以避免节点在极点附近聚集;最后求解了多个Dirichlet或Neumann边界条件下的Poisson-型方程算例。数值结果表明,谱元方法具有很高的精度。