On the convergence of a linearly implicit finite element method for the nonlinear Schrödinger equation

Journal article, 2024

We consider a model initial- and Dirichlet boundary–value problem for a nonlinear Schrödinger equation in two and three space dimensions. The solution to the problem is approximated by a conservative numerical method consisting of a standard conforming finite element space discretization and a second-order, linearly implicit time stepping, yielding approximations at the nodes and at the midpoints of a nonuniform partition of the time interval. We investigate the convergence of the method by deriving optimal-order error estimates in the 𝐿2 and the 𝐻1 norm, under certain assumptions on the partition of the time interval and avoiding the enforcement of a courant-friedrichs-lewy (CFL) condition between the space mesh size and the time step sizes.