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 $E{Q}_1^{\mathrm{rot}}$ element and zero-order Raviart–Thomas element for a generalized nonlinear Schrödinger equation. On the one hand, by introducing $\overrightarrow{p}=\mathrm{\nabla }u$ 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 $H^1$-norm and flux $\overrightarrow{p}=\mathrm{\nabla }u$ in $L^2$ 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 $O(h^2+{\tau }^2)$ 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.

Keywords:

• Generalized nonlinear Schrödinger equation
• nonconforming mixed finite element method
• semi-discrete and fully discrete schemes
• superclose
• superconvergence analysis

2000 AMS Subject Classifications:

• 65N15
• 65N30

Disclosure statement

No potential conflict of interest was reported by the author(s).