A Robust numerical approach for singularly perturbed time delayed parabolic partial differential equations

We consider the numerical solution of a initial boundary value problem with a time delay. The problem under consideration is singularly perturbed from the mathematical perspective. Assuming that the coefficients of the differential equation are smooth, we construct and analyze the finite difference method whose solutions converge pointwise at all points of the domain independently of the singular perturbation parameter. The method permits its extension to the case of adaptive meshes, which may be used to improve the solution. Numerical examples are presented to demonstrate the effectiveness of the method. The convergence obtained in practice satisfies the theoretical predictions.