Linearly Implicit Finite Element Methods Approximating the Solution to the Nonlinear Schrödinger Equation with a...

We consider an initial- and Dirichlet boundary- value problem for a nonlinear Schrödinger equation of the form $u_t=\text{i}\Delta u+\text{i}Vu+\text{i}\mu {\left|u\right|}^{\beta }u+f$ over $[0,T]\times \Omega$ , where $T>0$ , $\Omega \subset R^d$ for $d\in \{1,2,3\}$ , $\beta \in (0,1)$ , V is a real-valued time-independent potential and $\mu$ is a nonzero real number. The solution to the problem is approximated by the Linearized Backward Euler finite element (LBEFE) method which is dissipative and the Linearized Crank-Nicolson finite element (LCNFE) one which is conservative. Letting $\tau$ be the time-step and h be the width of the finite element partition of the space domain, we provide an optimal order $O(\tau +h^2)$ error estimate in the $L^2$ norm for both methods, and an $O({\tau }^{\alpha }+h)$ error estimate in the $H^1$ norm, where $\alpha =\frac{3}{4}$ in the (LBEFE) method and $\alpha =\frac{1}{2}$ in the (LCNFE) one. For $d=1$ , no CFL conditions are imposed, while for $d=2$ or 3, a mesh condition of the form $(h^{\frac{2-d}{2}}{|ln\left(h\right)|}^{\frac{d-1}{d}}{\tau }^{\alpha }+h^{2-\frac{d}{2}})=O\left(1\right)$ is required. Finally, with results from numerical experiments, we investigate the performance of the methods proposed and analyzed.