Superconvergence analysis of a MFEM for BBM equation with a stable scheme. (1st July 2021) - Wang, Junjun; Li, Meng; Jiang, Mengping
Abstract: In this paper, superconvergence properties are presented for the nonlinear Benjamin-Bona-Mahony (BBM) equation with a mixed finite element method (MFEM). We propose an Euler scheme combined with the artificial Douglas-Dupont regularization terms, which guarantee the stability of the numerical scheme. Splitting technique is utilized to get rid of the ratio between the time step τ and the subdivision parameter h . Temporal error estimates and unconditional spatial error estimates are gained through some techniques, such as the function's monotonicity and the Green formula and so on. In turn, the regularities of the solutions about the time-discrete equations and the boundedness of the numerical solution are derived. Based on the above achievements, the unconditional superconvergent results of u n in H 1 -norm and q → n in L 2 -norm with order O ( h 2 + τ ) are obtained through the trigonometric inequality. The global superconvergent results are deduced by use of the interpolated postprocessing operators. Numerical example shows the validity of the theoretical analysis.