Numerical solution of two-dimensional nonlinear Schrödinger equation using a new two-grid finite element method. (English) Zbl 1434.65185
Summary: A new two-grid finite element scheme is presented for two-dimensional nonlinear Schrödinger equation. One Newton iteration is applied on the fine grid to linearize the nonlinear system using the coarse-grid solution as the initial guess, and furthermore one more linear system on the coarse space is solved. The error estimations of the two-grid solution in the \(L^2\) and \(H^1\) norm are given. It is shown, both theoretically and numerically, that the coarse space can be extremely coarse and two-grid algorithm still achieves optimal approximation as long as the mesh sizes satisfy \(H = O(h^{\frac{1}{3}})\).
MSC:
| 65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |
| 65M55 | Multigrid methods; domain decomposition for initial value and initial-boundary value problems involving PDEs |
| 65H10 | Numerical computation of solutions to systems of equations |
| 65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
| 35Q41 | Time-dependent Schrödinger equations and Dirac equations |
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