Numerical analysis of a system of semilinear singularly perturbed first‐order differential equations on an adaptive grid

Abstract

An adaptive grid method based on the backward Euler formula for a system of semilinear singularly perturbed initial value problems is studied. Based on the a priori error analysis and mesh equidistribution principle, we prove that the convergence rate of our semidiscrete adaptive grid method is first order, which is robust with respect to the perturbation parameters. Then, in order to construct a fully discrete adaptive grid method, a standard residual‐type a posterior error estimation is constructed by using the linear polynomial interpolation technique. A partly heuristic argument based on this a posteriori error estimator leads to an optimal monitor function, which is used to design an adaptive grid algorithm. Furthermore, we also extend our presented adaptive grid method to a nonlinear system of singularly perturbed problem arising in the modeling of enzyme kinetics and a system of singularly perturbed delay differential equations, respectively. Finally, numerical results are provided to illustrate the effectiveness of our presented adaptive grid method.