Superconvergence analysis of a nonconforming MFEM for nonlinear Schrödinger equation

Abstract

In this paper, an efficient nonconforming mixed finite element method (MFEM) is studied with element and zero-order Raviart–Thomas element for a generalized nonlinear Schrödinger equation. On the one hand, by introducing as an intermediate variable, we split the equation into two low-order equations and present a semi-discrete scheme for nonconforming MFEM and prove its existence and uniqueness. And the superconvergence results for original variable u in broken -norm and flux in norm are derived by using the proved properties of the above nonconforming MFEs. On the other hand, a linearized Crank–Nicolson fully discrete scheme is constructed and the superclose and superconvergence results with order for above variables are also derived. The keys to our analysis are the following two skills: one is an important novel property for the above MFEs (see Lemma 2.3) and another is a splitting technique for nonlinear terms, while previous literature always only obtain the convergent estimates in a routine way. Finally, two numerical examples are provided to confirm the theoretical analysis. Here, h is the subdivision parameter and τ is the time step.