A relaxation-type Galerkin FEM for nonlinear fractional Schrödinger equations

Abstract

In this paper, a class of nonlinear Riesz space-fractional Schrödinger equations are considered. Based on the standard Galerkin finite element method in space and a relaxation-type difference method in time, a fully discrete system is constructed. This scheme avoids solving the nonlinear systems and preserves the mass and energy very well. By the Brouwer fixed-pointed theorem, the unique solvability of the discrete system is proved. Moreover, we focus on a rigorous analysis of the optimal convergence properties for the fully discrete system. Finally, some numerical examples are given to validate the theoretical analysis.

Appendices

Appendix I: the two implicit FEMs

The first one is the implicit midpoint finite element method (IMFEM), which is to find $u_{h}^{n + 1}\in S_{h}$, such that

$$ i(\delta_{t} u_{h}^{n+\frac{1}{2}}, v_{h})-B(\bar{u}_{h}^{n+\frac{1}{2}}, v_{h})+\lambda(|\bar{u}_{h}^{n+\frac{1}{2}}|^{2}\bar{u}_{h}^{n+\frac{1}{2}}, v_{h})= 0, ~~v_{h}\in S_{h}, $$

(1)

with the initial condition ${u_{h}^{0}}=I_{h}u_{0}$. To embark the numerical experiments, we develop following efficient iterative scheme as follows:

$$ (\mathcal{H}^{(s + 1)}. v_{h})-({u_{h}^{n}}, v_{h})+i\frac{\tau}{2}B(\mathcal{H}^{(s + 1)}, v_{h})-i\frac{\lambda\tau}{2}(|\mathcal{H}^{(s)}|^{2}\mathcal{H}^{(s)}, v_{h}) = 0. $$

(2)

The initial iterative is selected as, for n ≥ 1,

$$\mathcal{H}^{(0)}=\frac{3{u_{h}^{n}}-u_{h}^{n-1}}{2}, $$

and for n = 0, we find $\mathcal {H}^{(0)}$ by the following:

$$(\mathcal{H}^{(0)}, v_{h})-({u_{h}^{0}}, v_{h})+i\frac{\tau}{2}B({u_{h}^{0}}, v_{h})-i\frac{\lambda\tau}{2}(|{u_{h}^{0}}|^{2}{u_{h}^{0}}, v_{h})= 0. $$

Once we obtain $\mathcal {H}^{(s + 1)}$ by (111), then $\bar {u}_{h}^{n + 1/2}$ is reached if $\mathcal {H}^{(s + 1)}$ converges. In the end, we can get $u_{h}^{n + 1}$ by $u_{h}^{n + 1}= 2\bar {u}_{h}^{n + 1/2}-{u_{h}^{n}}$.

The second one is the Crank-Nicolson finite element method (CNFEM), which is to find $u_{h}^{n + 1}\in S_{h}$, such that

$$ i(\delta_{t}u_{h}^{n+\frac{1}{2}}, v_{h})-B(\bar{u}_{h}^{n+\frac{1}{2}},v_{h})+\frac{\lambda}{2}\left( \left( |u_{h}^{n + 1}|^{2}+|{u_{h}^{n}}|^{2}\right)\bar{u}_{h}^{n+\frac{1}{2}}, v_{h}\right)= 0,~~\forall v_{h}\in S_{h}, $$

(3)

with the initial condition ${u_{h}^{0}}=I_{h}u_{0}$. We propose an iteration procedure as follows:

$$ i\left( \frac{u_{h}^{n + 1(s + 1)}-{u_{h}^{n}}}{\tau}, v_{h}\right)-\frac{1}{2}B(\bar{u}_{h}^{n + 1(s + 1)}+{u_{h}^{n}}, v_{h})+(H^{n + 1(s)}, v_{h})= 0, $$

(4)

with the boundary condition

$$ u_{h}^{n + 1(s + 1)}= 0,~~on~\partial{\Omega}, $$

(5)

where

$$ H^{n + 1(s)}=\frac{\lambda}{4}(|u_{h}^{n + 1(s)}|^{2}+|{u_{h}^{n}}|^{2})(u_{h}^{n + 1(s)}+{u_{h}^{n}}), $$

(6)

and

$$ u_{h}^{n + 1(0)}=\left\{ \begin{array}{lll} &{u_{h}^{n}},&n = 0,\\ &2{u_{h}^{n}}-u_{h}^{n-1},~~&n\geq 1. \end{array} \right. $$

(7)

Appendix II: the proof of Lemma 8

First, we introduce the following system as follows:

$$ \left\{ \begin{array}{lll} & -\frac{\partial^{\alpha} \omega}{\partial |x|^{\alpha}}=f, \hspace{0.5cm} f\in L^{2}({\Omega}),\\ & \omega(a)=\omega(b)= 0 . \end{array} \right. $$

(8)

Obviously, it exists a unique solution $u\in H_{0}^{\frac {\alpha }{2}}\cap H^{\alpha }({\Omega })$, such that

$$\begin{array}{@{}rcl@{}} && \|\omega\|_{\alpha}\leq C\|f\|, \quad \alpha\neq \frac{3}{2}; \end{array} $$

(9)

$$\begin{array}{@{}rcl@{}} && \|\omega\|_{\alpha-\sigma}\leq C\|f\|, \quad \alpha\neq \frac{3}{2}, \quad 0<\sigma<\frac{1}{2}. \end{array} $$

(10)

Let $u\in H_{0}^{\frac {\alpha }{2}}\cap H^{\eta }({\Omega })$, α/2 < η ≤ r + 1. Then, by (17) and (57), we obtain the following:

$$ \|u-P_{h}u\|_{\frac{\alpha}{2}}^{2}\!\leq\! CB(u-P_{h}u, u-P_{h}u) = CB(u-P_{h}u, u-I_{h}u)\!\leq\! C\|u-P_{h}u\|_{\frac{\alpha}{2}}\|u-I_{h}u\|_{\frac{\alpha}{2}}. $$

(11)

Therefore, we have as follows:

$$ \|u-P_{h}u\|_{\frac{\alpha}{2}}\leq C\|u-I_{h}u\|_{\frac{\alpha}{2}}. $$

(12)

Thanks to the approximation property [41], we derive with the following:

$$ \|u-P_{h}u\|_{\frac{\alpha}{2}}\leq Ch^{\eta-\frac{\alpha}{2}}\|u\|_{\eta}. $$

(13)

For α≠ 3/2, it follows from (17) and (117) that

$$\begin{array}{@{}rcl@{}} (u-P_{h}u, f) &=& B(u-P_{h}u, \omega) \\ &=& B(u-P_{h}u, \omega-I_{h}\omega) \\ & \leq& C\|u-P_{h}u\|_{\frac{\alpha}{2}}\|\omega-I_{h}\omega\|_{\frac{\alpha}{2}} \\ &\leq & Ch^{\eta}\|u\|_{\eta}\|\omega\|_{\alpha}\leq Ch^{\eta}\|u\|_{\eta}\|f\|_{0}. \end{array} $$

(14)

Thus, we have as follows:

$$ \|u-P_{h}u\|\leq Ch^{\eta}\|u\|_{\eta}. $$

(15)

Similarly, for α = 3/2, it follows that

$$ \|u-P_{h}u\|\leq Ch^{\eta-\sigma}\|u\|_{\eta}, \quad 0<\sigma<\frac{1}{2}. $$

(16)

Therefore, we have completed the proof of Lemma 8.