Combined finite volume element method for singularly perturbed reaction–diffusion problems

A finite volume element method is presented for solving singularly perturbed reaction–diffusion problems in two-dimensional domain. The concept of the finite volume method is used to discretize the unsteady scalar reaction–diffusion equation, while the finite element method is applied to estimate the gradient quantities at cell faces. The error estimate in a discrete H01 norm for the approximate solution of one-dimensional stationary reaction–diffusion equation is also proved. Numerical solutions of the singularly perturbed reaction–diffusion problems on two-dimensional uniformly rectangular grids are presented to demonstrate the robustness and efficiency of the combined method. The numerical solutions demonstrate that the combined method is stable and provides accurate solution without spurious oscillation along the high-gradient boundary layers.