In this article, an energy dissipation property preserving modified Crank–Nicolson (CN) fully-discrete scheme is developed and investigated with the nonconforming quadrilateral Quasi-Wilson element for nonlinear Benjamin–Bona–Mahony–Burgers (BBMB) equation. Different from the so-called popular splitting technique used in the previous studies, the boundedness of numerical solution in the broken \(H^1\) -norm is proved directly, which leads to the existence and uniqueness of the numerical solution can be testified strictly via the Brouwer fixed point theorem. Then, with the help of the special character of this element, that is, the consistency error can be estimated with order \(O(h^2),\) one order higher than its interpolation error, the unconditional supercloseness of order \(O(h^2+\tau ^2)\) on quadrilateral meshes are deduced rigorously without any restriction between mesh size h and time step \(\tau .\) Further, through the interpolation post-processing technique, the unconditional global superconvergence estimate on quadrilateral meshes is acquired. Finally, some numerical experiments are conducted to confirm the theoretical analysis. It is shown that the proposed scheme has much better performance than the famous Wilson element. Here, we mention that the presented approach and analysis are also valid to some other known low order conforming and nonconforming finite elements.
