Unconditional superconvergence analysis of nonconforming [formula omitted] finite element method for the nonlinear coupled predator-prey equations.

A nonconforming E Q 1 r o t finite element method (FEM) is studied for the nonlinear coupled predator-prey equations. The superconvergence estimates are derived for the semi-discrete and Crank-Nicolson (C-N) fully discrete schemes based on the two special properties of this element: one is that the interpolation operator is equivalent to its projection operator, and the other is that the consistency error can be estimated as order O (h 2) in the broken H 1 -norm when the exact solution belongs to H 3 (Ω) , which is one order higher than that of its interpolation error estimate. Moreover, the unique solvability of the nonlinear coupled semi-discrete scheme is certified through the Brouwer fixed point theorem analytically. On the other hand, the stability of the decoupled C-N fully discrete scheme is proved by mathematical induction, which leads to the unconditional superconvergence of order O (h 2 + τ 2) without the ratio between the time-step τ and the mesh size h. Finally, numerical examples are given to demonstrate the validity of our method. • Based on two properties of E Q 1 r o t element, we get superclose and superconvergence for the semi-discrete and C-N schemes. • The unique solvability of the nonlinear coupled semi-discrete scheme is proved by Brouwer fixed point theorem analytically. • The stability of the decoupled C-N scheme is proved by mathematical induction. • From the stability, we can eliminate the inverse inequality to get the superconvergence without time-step restriction. • The analysis and the results presented are also valid for other popular finite elements when they satisfy some conditions. [ABSTRACT FROM AUTHOR]