A robust adaptive grid method for first-order nonlinear singularly perturbed Fredholm integro-differential equations

In this paper, a robust adaptive grid method is developed for solving first-order nonlinear singularly perturbed Fredholm integro-differential equations (SPFIDEs). Firstly such SPFIDEs are discretized by the backward Euler formula for differential part and the composite numerical quadrature rule for integral part. Then both a prior and an a posterior error analysis in the maximum norm are derived. Based on the prior error bound and the mesh equidistribution principle, it is proved that there exists a mesh gives optimal first-order convergence which is robust with respect to the perturbation parameter. Finally, the posterior error bound is used to choose a suitable monitor function and design a corresponding adaptive grid generation algorithm. Numerical results are given to illustrate our theoretical result.