Superconvergence analysis of two-grid methods for bacteria equations

In this paper, two-grid methods (TGMs) are developed for a system of reaction-diffusion equations of bacterial infection with initial and boundary conditions. The backward Euler (B-E) and Crank–Nicolson (C-N) fully discrete schemes are established, and the existence and uniqueness of the solutions of these schemes are proved. Moreover, based on the combination technique of the interpolation and Ritz projection and derivative transfer trick which are important ingredients in the TGMs, the superclose estimates of order O ( h 2 + H 4 + τ ) and O ( h 2 + H 4 + τ 2 ) in H 1 -norm are deduced for the above schemes, respectively, where h is fine mesh size, H is coarse mesh size, and τ is time step size. Then, by the interpolated postprocessing approach, the corresponding global superconvergence results are obtained. Finally, some other popular finite elements are discussed and numerical results are provided to verify the theoretical analysis, which show that the computing cost of TGMs are only half of Galerkin finite element methods (FEMs) for the test problem.