奇异摄动周期边界问题的自适应计算方法

Adaptive numerical method for singularly perturbed equation with periodical boundary value problem

针对椭圆型奇异摄动周期边界方程,提出有效的计算方法,并证明所构造的计算方法是自适应的,随着小参数的变小,网格剖分数目不需要很大,仍可以得到很好的计算效果.讨论边界层的性质,将解的奇性分离为光滑部分和奇性部分,对光滑部分和奇性部分的各阶偏导数进行估计;在Shishkin网格上提出有限差分方法,证明离散极值原理和一致稳定性;构造相应的闸函数以证明所提方法具有一致收敛性.给出一个数值例子,计算结果表明,计算方法拟合了边界层的性质,也说明理论分析的正确性.所提出的计算方法可应用于类似奇异摄动问题的计算.

Aimed at singularly perturbed elliptic partial differential equation with periodical boundary value problem,an effective numerical method was constructed.Adaptive convergence was proved.Effective computational result was obtained with few number grid when the parameter was small.Firstly,the property of boundary layer was discussed.The solution was decomposed into the smooth component and the singular component.The derivatives of the smooth component and the singular component were estimated.Secondly,finite difference method was proposed on the Shishkin mesh.The discrete maximum principle and the uniform stability result were studied.Thirdly,uniform convergence is proved by constructing the barrier function.Finally,numerical experiment was proposed to support the theoretical result.Numerical results show that the presented method fitted the property of boundary layer well.The solution for this kind of multi-scale problem was provided in theory.The presented numerical method can be applied to calculate other singularly perturbed problems.