Maximum norm a posteriori error estimates for a singularly perturbed differential difference equation with small delay

•We consider an adaptive grid approach to solve a singularly perturbed differential difference equation with small delay.•The mesh is constructed adaptively by equidistributing a arc-length monitor function.•We derive a first-order maximum norm a posteriori estimate for the full discretization scheme.•A first-order rate of convergence, independent of the perturbation parameter and the small delay parameter, is established.

A singularly perturbed differential difference equation with small delay is discretized on an adaptive grid which is formed by equidistributing arc-length monitor function. We first derive first-order maximum norm a posteriori estimates for the full discretization scheme of these problems. Then a first-order rate of convergence, independent of the perturbation parameter and the small delay parameter, is established. Numerical results are provided that support our theoretical estimates.