Unconditional superconvergence analysis of a new energy stable nonconforming BDF2 mixed finite element method for BBM–Burgers equation

Abstract

The focus of this paper is to establish a new energy stable 2-step backward differentiation formula (BDF2) fully-discrete mixed finite element method (MFEM) for the Benjamin–Bona–Mahony–Burgers (BBM–Burgers) equation with the nonconforming rectangular $E{Q}_1^{rot}$ element and zero-order Raviart–Thomas (R–T) element ($E{Q}_1^{rot}/Q_{10}\times Q_{01}$). Based on the energy stable property, the existence and uniqueness of the numerical solution are proved by the Brouwer fixed point theorem. Subsequently, with the assistance of some typical properties of this element pair and the interpolation post-processing approach, the unconditional superclose and superconvergence results with $O(h^2+{\tau }^2)$ are obtained directly without any constraints between the spatial partition parameter $h$ and the time step $\tau$. It is worthy to mention that the method and analysis presented herein are very different from the time–space splitting approach utilized in the previous studies and simplify the implement. Finally, two numerical experiments are executed to validate the theoretical analysis.