Unconditional optimal error estimates of linearized, decoupled and conservative Galerkin FEMs for the Klein-Gordon-Schrödinger equation. (English) Zbl 1476.65259
Summary: This paper is concerned with unconditionally optimal error estimates of linearized leap-frog Galerkin finite element methods (FEMs) to numerically solve the \(d\)-dimensional \((d=2,3)\) nonlinear Klein-Gordon-Schrödinger (KGS) equation. The proposed FEMs not only conserve the mass and energy in the given discrete norm but also are efficient in implementation because only two linear systems need to be solved at each time step. Meanwhile, an optimal error estimate for the proposed methods is derived by using the temporal-spatial error splitting techniques, which split the error between the exact solution and the numerical solution into two parts, i.e., the temporal error and the spatial error. Since the spatial error is \(\tau\)-independent, the boundedness of the numerical solution in \(L^\infty\)-norm follows an inverse inequality immediately without any restriction on the grid ratios. Then, the optimal \(L^2\) error estimates for \(r\)-order FEMs are derived unconditionally. Numerical results in both two and three dimensional spaces are given to confirm the theoretical predictions and demonstrate the efficiency of the methods.
MSC:
| 65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
| 65M15 | Error bounds for initial value and initial-boundary value problems involving PDEs |
| 65N15 | Error bounds for boundary value problems involving PDEs |
| 65N12 | Stability and convergence of numerical methods for boundary value problems involving PDEs |
| 35Q55 | NLS equations (nonlinear Schrödinger equations) |
| 35Q41 | Time-dependent Schrödinger equations and Dirac equations |
Keywords:
Klein-Gordon-Schrödinger equations; finite element method; optimal error estimates; linearized method; conservative schemes; unconditional convergenceSoftware:
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