自适应网格下齐次奇异摄动问题的一致收敛性分析

Uniform convergence analysis of homogenous singular perturbation problem with adaptive grids

利用自适应移动网格方法求解齐次奇异摄动边值问题,通过常数与解的一阶导数幂的线性组合构造控制函数来进行网格自适应,分析齐次奇异摄动边值问题在此自适应非均匀网格上的收敛性.利用极值原理证明离散问题解的存在唯一性.对离散问题的数值解及其分段线性插值进行误差估计,分别得到一个与ε无关的一阶误差界.有效地解决齐次奇异摄动方程难以在均匀网格上进行数值求解的问题.数值算例验证理论分析的有效性.

A homogenous singular perturbation boundary value problem was solved numerically by using the adaptive-moving mesh method,and the adaptive grid was suggested according to a control function,which was a linear combination of a constant floor and a power of the first derivative of the solution.The convergence of the homogenous singular perturbation boundary value problem was analyzed with this adaptive grid.The extremum principle was used to prove the existence and uniqueness of the solutions of its discrete problem.The error of the numerical solution and the piecewise linear interpolation for the numerical solution of the discrete problem were estimated.Finally,a first-order error bound independent of the ε was derived respectively.It was difficult to solve the singular perturbation problem with the uniform grids,but by using the method presented this difficulty was effectively solved.It was shown by a numerical example that the theoretical analysis was valid.