Unconditional superconvergence analysis of an energy conservation scheme with Galerkin FEM for nonlinear Benjamin-Bona-Mahony equation

In this paper, a general energy conservation Crank-Nicolson (CN) fully-discrete finite element method (FEM) scheme is developed for solving the nonlinear Benjamin-Bona-Mahony (BBM) equation, and the unconditional supercloseness and superconvergence behavior are discussed with the nonconforming EQ(1)(rot) element. Different from the time-space splitting technique, the boundedness of numerical solution in the broken H-1-norm is obtained directly based on the above energy conservation property, and the well-posedness of numerical solution is proved rigorously through the Brouwer fixed point theorem. Then, with the help of the special characters of this element and the interpolation post-processing technique, the unconditional superclose and superconvergence results are deduced strictly without any restriction between mesh size h and time step tau. Further, our analysis is extended to some other popular conforming and nonconforming finite elements. Finally, two numerical experiments are implemented to confirm the theoretical analysis. It is worth mentioning that this paper not only improve the previous results, but also can be regard as a framework for solving the BBM equation and some other PDEs with Galerkin FEMs.