Hamilton系统的有限元研究

Study of Finite Elements for Hamilton Systems

该文对Hamilton系统的连续有限元法证明了两个优美的性质:在任何情形m次有限元总是能量守恒的,它对线性系统也是辛的,且对非线性系统每次步进是高精度O(h^(2m+1))近似辛的.在长时间计算中时空平面上轨道和周期的偏离随时间线性增长.数值实验表明其偏离比其他算法小.

Two nice properties of the continuous finite element method for Hamilton systems are proved as follows：in any case the m-degree finite elements always preserve the energy which is sympletic for linear systems and is approximately sympletic with high accuracy O（h^（2m＋1）） in each stepping for nonlinear systems.In long-time computation the deviation of trajectories and their periods in time-space plane will crease linearly with time.Numerical experiments show that their deviations are often smaller than that of other schemes.