Parameter-uniform finite element method for two-parameter singularly perturbed parabolic reaction-diffusion problems

In this paper, parameter-uniform numerical methods for a class of singularly perturbed one-dimensional parabolic reaction-diffusion problems with two small parameters on a rectangular domain are studied. Parameter-explicit theoretical bounds on the derivatives of the solutions are derived. The method comprises a standard implicit finite difference scheme to discretize in temporal direction on a uniform mesh by means of Rothe's method and finite element method in spatial direction on a piecewise uniform mesh of Shishkin type. The method is shown to be unconditionally stable and accurate of order O(N ^-2(ln N)² + Δt). Numerical results are given to illustrate the parameter-uniform convergence of the numerical approximations.