An efficient adaptive grid method for a system of singularly perturbed convection-diffusion problems with Robin boundary conditions

A system of singularly perturbed convection-diffusion equations with Robin boundary conditions is considered on the interval [0,1]. It is shown that any solution of such a problem can be expressed to a system of first-order singularly perturbed initial value problem, which is discretized by the backward Euler formula on an arbitrary nonuniform mesh. An a posteriori error estimation in maximum norm is derived to design an adaptive grid generation algorithm. Besides, in order to establish the initial values of the original problems, we construct a nonlinear optimization problem, which is solved by the Nelder-Mead simplex method. Numerical results are given to demonstrate the performance of the presented method.