Superconvergence analysis for nonlinear parabolic equation with $EQ_1^\mathrm{{rot}}$ nonconforming finite element

Abstract

$EQ_1^\mathrm{{rot}}$ nonconforming finite element method (FEM) applied to a class of nonlinear parabolic equation is discussed. First, by use of two typical characters of this element (one is that the associated FE interpolation operator is identical to its traditional Ritz projection operator; the other is that the consistency error is of order $O(h^2)$, one order higher than the interpolation error, when the exact solution of the problems belongs to $H^3(\Omega )$), the supercloseness of order $O(h^2)$ in broken $H^1$-norm for semi-discrete scheme is obtained without the boundedness of numerical solution in $L^\infty$-norm through splitting the numerical solution into several parts. Moreover, we get the desired result with requirement of $u,u_t\in H^3(\Omega )$ only. Secondly, a linearized Crank–Nicolson fully discrete scheme is proposed and the superclose property of order $O(h^2+\tau ^2)$ is derived by constructing a suitable auxiliary problem. Finally, a numerical example is carried out to confirm our theoretical analysis. Here, h is the subdivision parameter and $\tau$ is the time step.