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

Abstract

A nonconforming EQ1rot 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(h2) in the broken H1-norm when the exact solution belongs to H3(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(h2+τ2) 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.