ε-Uniformly convergent numerical scheme for singularly perturbed delay parabolic partial differential equations

This paper studies the numerical solutions of singularly perturbed parabolic convection–diffusion problems with a delay in time. We divide the domain using a piecewise uniform adaptive mesh in the spatial direction and a uniform mesh in the temporal direction. Further, we discretize the time derivative by the backward-Euler scheme and the spatial derivatives by the upwind finite difference scheme. We obtain the maximum principle and carry out the stability analysis. Then we prove that the proposed scheme is ε-uniform convergence of first-order in time and first-order up to a logarithmic factor in space. Numerical results are carried out to verify the theoretical results.