Analysis of finite element two-grid algorithms for two-dimensional nonlinear Schrödinger equation with wave operator

Two-grid algorithms based on two conservative and implicit finite element methods are studied for two-dimensional nonlinear Schrödinger equation with wave operator. The existence and uniqueness of their solutions and the conservative laws are established for both schemes. To linearize the fully discrete nonlinear coupling problem, the original problem is decomposed into two equivalent nonlinear coupled hyperbolic–parabolic equations. Algorithm 1 has three steps based on one Newton iteration on the fine grid and further correction on coarse grid, while Algorithm 2 has three steps based on two Newton iterations on the fine grid. Optimal order L p error estimations of the two-grid algorithms are conducted in detail by optimal order L p error estimates of finite element methods without any time-step size conditions. Both theoretically and numerically are shown that the coarse space can be extremely coarse, with no loss in the order of accuracy, and the two-grid algorithms still achieve the optimal convergence order when the mesh sizes satisfy H = O ( h 1 3 ) for Algorithm 1 and H = O ( h 1 4 ) for Algorithm 2.