A parameter-robust finite difference method for singularly perturbed delay parabolic partial differential equations

Abstract

A Dirichlet boundary value problem for a delay parabolic differential equation is studied on a rectangular domain in the x - t plane. The second-order space derivative is multiplied by a small singular perturbation parameter, which gives rise to parabolic boundary layers on the two lateral sides of the rectangle. A numerical method comprising a standard finite difference operator (centred in space, implicit in time) on a rectangular piecewise uniform fitted mesh of N x × N t elements condensing in the boundary layers is proved to be robust with respect to the small parameter, or parameter-uniform, in the sense that its numerical solutions converge in the maximum norm to the exact solution uniformly well for all values of the parameter in the half-open interval ( 0 , 1 ] . More specifically, it is shown that the errors are bounded in the maximum norm by C ( N x - 2 ln 2 N x + N t - 1 ) , where C is a constant independent not only of N x and N t but also of the small parameter. Numerical results are presented, which validate numerically this theoretical result and show that a numerical method consisting of the standard finite difference operator on a uniform mesh of N x × N t elements is not parameter-robust.