Layer adapted finite element method for two-parameter singularly perturbed delay parabolic problems

A layer-adapted finite element method is applied to solve a singularly perturbed delay parabolic differential equation with two perturbation parameters. The numerical method involves two steps of discretization in the decoupled approach. First, the temporal variable is discretized uniformly and approximated by a finite element piecewise linear continuous function. The time derivative of this approximation in each mesh is constant in the time direction. It is a function of the space variable. Then, the time derivative term in the equation is approximated by the equivalent Crank-Nicholson finite difference method. Next, for the semi-discretized problem, a fitted mesh (layer-adapted) finite element method is applied using the Shishkin mesh for the spatial discretization. The solution in the spatial elements is approximated by piecewise quadratic polynomial functions. Finally, the finite element solution is presented as a linear combination of the piecewise linear temporal FE approximate functions and the piecewise quadratic spatial FE approximate functions. That is, a piece wise smooth continuous function in the given domain, which empowers finding an approximation solution at any desired point of the domain. It is proven that the numerical method is convergent with the second order of convergence in the temporal direction and with almost the second order of convergence in the spatial direction. Using the MATLAB software, some numerical experiments are performed to verify the applicability of the numerical method.