无界区域抛物方程自然边界元方法

摘要

本文应用自然边界元方法求解无界区域抛物型初边值问题.首先将控制方程对时间进行离散化,得到关于时间步长离散化的椭圆型问题.通过Fourier展开,导出相应问题的自然积分方程和Poisson积分公式.研究了自然积分算子的性质,并讨论了自然积分方程的数值解法,最后给出数值例子.从而解决了抛物型问题的自然边界归化和自然边界元方法.%In this paper we introduce an implementation for the etfficient numerical solution of exterior initial boundary value problem for parabolic equation. The problem is reformulated as an equivalent one on a boundary Γ using natural boundary reduction.The governing equation is first discretized in tine, leading to a time-stepping scheme,where an exterior elliptic problem has to be solved in each time step. By Fourier expansion, we derive a natural integral equation of the elliptic problem related to time step and Poisson integral integral formula over exterior circular domain. Finite element discretization of the natural integral equation is employed to solve this problem. The computational aspects of this method are discussed. Numerical results are presented to illustrate feasibility and efficiency of our method.