Unconditional superconvergence of the fully-discrete schemes for nonlinear prey-predator model

Abstract

In this paper, the linearized Backward-Euler (BE) and backward-differential-formula (BDF) schemes are developed for the nonlinear and coupled prey-predator system with finite element method (FEM). The stability of the discrete solution in H1-norm is proved and used to deal with the quadratic term appeared in the system. Thanks to the boundness of the discrete solution obtained from the stability, the superclose results without time-step restriction are derived, in which a novel splitting technique is adopted to cope with the estimations related to the discrete time derivatives. Then, based on the high accuracy analysis, the unconditional superconvergence results of order O(h2+τ) and O(h2+τ2) in H1-norm are derived for BE and BDF schemes through post-processing approach, respectively. Here, h is the subdivision parameter and τ, the time step.