A linearized time stepping scheme for finite elements applied to Gray-Scott model

Abstract

This paper describes a numerical scheme for solving a reaction-diffusion system, specifically the Gray-Scott model. The scheme is a two-step process, combining the Crank-Nicolson method in the first step with the second-order backward differentiation formula in the second step. This combination ensures unconditional stability in both $L^2$ and $H^1$-norms and allows for optimal error estimates. The scheme's efficiency is further enhanced by a temporal-spatial error splitting technique, enabling the derivation of error estimates without any restrictions on meshes size steps. By dividing the error into temporal and spatial components, the unconditional convergence result is deduced. The regularity of a time-discrete system is established through the analysis of the temporal error. Additionally, the classical Ritz projection is used to achieve the optimal spatial error in the $L^2$-norm, which is crucial for eliminating the constraint of Δt. Since the spatial error is not dependent on the time step, the boundedness of the numerical solution in $L^{\infty }$-norm follows an inverse inequality immediately without any restriction on the grid mesh.