A New Error Analysis of Nonconforming $EQ^{rot}_1$ FEM for Nonlinear BBM Equation (Advances in Applied Mathematics and Mechanics, 2018)
Nonconforming $EQ^{rot}_1$ element is applied to solving a kind of nonlinearBenjamin-Bona-Mahony (BBM for short) equation both for space-discrete and fullydiscrete schemes. A new important estimate is proved, which improves the result ofprevious works with the exact solution $u$ belonging to $H^2(Ω)$ instead of $H^3(Ω)$. Andthen, the supercloseness and global superconvergence estimates in broken $H^1$ normare obtained for space-discrete scheme. Further, the superclose estimates are deducedfor backward Euler and Crank-Nicolson schemes. To confirm our theoretical analysis,numerical experiments for backward Euler scheme are executed. It seems that the resultspresented herein have never been seen for nonconforming FEMs in the existingliterature.