线性奇异摄动两点边值问题的自适应解法

Adaptive Solution of Singularly Perturbed Linear Two-Point Boundary Value Problems

摘要: 本文讨论非守恒型线性奇异摄动两点边值问题的基于弧长等分布原理的自适应差分解法及其误差分析。引入对应的线性算子的伴随算子，利用伴随算子的格林函数的性质，证明了关于线性算子的稳定性。引入二次插值函数，可得在任意网格上提出的差分格式的后验误差估计。在均匀分布数值解弧长的自适应网格上，将引入的伴随线性算子离散化，利用离散格林函数的性质，证明了原问题数值解弧长的有界性，又由后验误差估计结论，最终可以证明原问题基于弧长等分布原理的自适应差分格式关于小摄动参数的一阶的一致收敛性。文章最后进行了数值实验，数值结果表明了误差分析的正确性和方法的可行性。

Abstract: In this paper, an adaptive difference method based on the arc-length equal distribution principle and its error analysis for non conservative linear singularly perturbed two-point boundary value problems are discussed. The adjoint operator of the corresponding linear operator is introduced. By using the properties of the Green’s function of the adjoint operator, the stability of the linear opera-tor is proved. By introducing the quadratic interpolation function, the posterior error estimates of the difference schemes proposed on arbitrary grids can be obtained. On the adaptive grid with uni-form arc length, the introduced adjoint linear operator is discretized, and the boundedness of the arc length of the numerical solution of the original problem is proved by using the properties of the discrete Green’s function. From the conclusion of a posteriori error estimation, the first-order uni-form convergence of the adaptive difference scheme for the original problem based on the principle of equal arc length distribution with respect to small perturbation parameters can be proved. Fi-nally, numerical experiments are carried out. The numerical results show that the error analysis is correct and the method is feasible.