Superconvergence and extrapolation of a second-order θ fully discrete scheme with nonconforming FEM for degenerate nonlocal thermistor problems

Abstract

The main purpose of this paper is to study the superconvergence and extrapolation of nonconforming finite element method (FEM) for the degenerate nonlocal thermistor problem. Firstly, a new second-order $\theta$ fully discrete scheme is proposed with the nonconforming quasi-Wilson element on quadrilateral meshes. Notably, the scheme is reduced to the 2-step backward differential formula (BDF2) scheme if $\theta =0$, and is simplified as the Crank–Nicolson (CN) scheme when $\theta =\frac{1}{2}$. Secondly, the superclose and global superconvergence estimates with $O(h^2+{\tau }^2)$ are derived in the broken $H^1$-norm. Subsequently, the extrapolation result with $O(h^3+{\tau }^2)$ in the broken $H^1$-norm is deduced on rectangular meshes by the higher order consistency error estimate of the quasi-Wilson element and the asymptotic expansion result of the bilinear element. Here $h$ is the subdivision parameter and $\tau$ is the time step. Finally, some numerical experiments are performed to test the validity of the theoretical analysis.