# A robust adaptive grid method for a nonlinear singularly perturbed differential equation with integral boundary condition

• Li-Bin Liu
• Guangqing Long
• Zhongdi Cen
Original Paper

## Abstract

In this paper, the numerical solution of a nonlinear first-order singularly perturbed differential equation with integral boundary condition is considered. The discrete method is generated by a backward Euler formula and the grid is obtained by equidistributing a monitor function based on arc-length. We first give a rigorous error analysis for the numerical method of this problem on a grid that is constructed adaptively from a knowledge of the exact solution. A first-order rate of convergence, independent of the perturbation parameter, is established. Then, an a posteriori error bound and the corresponding convergence result are derived for the presented numerical scheme on an adaptive grid, which is constructed adaptively from a discrete approximation of the exact solution. At last, numerical experiments are given to illustrate our theoretical results.

## Keywords

Singularly perturbed Adaptive grid method Integral boundary condition Monitor function

## Notes

### Funding information

This work is supported by the National Science Foundation of China (11761015), the National Natural Science Foundation Mathematics Tianyuan Foundation of China (11826211, 11826212), the Natural Science Foundation of Guangxi (2017GXNSFBA198183), the key project of Guangxi Natural Science Foundation (2017GXNSFDA198014, 2018JJD110012), and the Zhejiang Provincial Public Welfare Project of China (LGF19A010001).

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