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

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.