A moving mesh refinement based optimal accurate uniformly convergent computational method for a parabolic system of boundary layer originated reaction–diffusion problems with arbitrary small diffusion terms

Abstract

In this paper, a system of time dependent boundary layer originated reaction dominated problems with diffusion parameters of different magnitudes, is considered for numerical analysis. The presence of these parameters lead to the boundary layer phenomena. Here, an optimal order uniformly accurate boundary layer adaptive method moving mesh method is proposed. This method is able to capture the layer phenomena without using a priori information of the solution. The problem is discretized by a modified implicit-Euler scheme in time direction. For the present system, adaptive mesh generation is required in space due to the singularly perturbed nature of the problem. For this purpose, a positive error monitor function is used whose equidistribution will move the mesh points towards the boundary layers. Parameter uniform error estimates are derived to show that the convergence rate is optimal with respect to the problem discretization. Numerical experiments strongly verify the theoretical findings and confirm the efficiency and accuracy of the proposed method.