Superconvergence Analysis of a BDF-Galerkin FEM for the Nonlinear Klein-Gordon-Schrödinger Equations with Damping Mechanism (Journal of Computational Mathematics, 2022)
The focus of this paper is on a linearized backward differential formula (BDF) scheme with Galerkin FEM for the nonlinear Klein-Gordon-Schrödinger equations (KGSEs) with damping mechanism. Optimal error estimates and superconvergence results are proved without any time-step restriction condition for the proposed scheme. The proof consists of three ingredients. First, a temporal-spatial error splitting argument is employed to bound the numerical solution in certain strong norms. Second, optimal error estimates are derived through a novel splitting technique to deal with the time derivative and some sharp estimates to cope with the nonlinear terms. Third, by virtue of the relationship between the Ritz projection and the interpolation, as well as a so-called "lifting'' technique, the superconvergence behavior of order $O(h^2+τ^2)$ in $H^1$-norm for the original variables are deduced. Finally, a numerical experiment is conducted to confirm our theoretical analysis. Here, $h$ is the spatial subdivision parameter, and $τ$ is the time step.
Keywords: KGSEs with damping mechanism; Linearized BDF Galerkin FEM; Optimal error estimates; Superconvergence.