We consider an initial- and Dirichlet boundary- value problem for a nonlinear Schrödinger equation of the form \(u_t=\,\textrm{i}\,{\varDelta }u+\textrm{i}\,V\,u+\textrm{i}\,\mu \,|u|^{\beta }\,u+f\) over \([0,T]\times {\varOmega }\) , where \(T>0\) , \({\varOmega }\subset {\mathbb {R}}^d\) 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 \(\big (h^{\frac{2-d}{2}}\,|\ln (h)|^{\frac{d-1}{d}} \,\tau ^{\alpha }+h^{2-\frac{d}{2}}\big )=O(1)\) is required. Finally, with results from numerical experiments, we investigate the performance of the methods proposed and analyzed.
