On the convergence of a linearly implicit finite element method for the nonlinear Schrödinger equation

IF 2.6 2区数学 Q1 MATHEMATICS, APPLIED

Studies in Applied Mathematics Pub Date : 2024-07-25 DOI:10.1111/sapm.12743

Mohammad Asadzadeh, Georgios E. Zouraris

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引用次数: 0

Abstract

We consider a model initial- and Dirichlet boundary–value problem for a nonlinear Schrödinger equation in two and three space dimensions. The solution to the problem is approximated by a conservative numerical method consisting of a standard conforming finite element space discretization and a second-order, linearly implicit time stepping, yielding approximations at the nodes and at the midpoints of a nonuniform partition of the time interval. We investigate the convergence of the method by deriving optimal-order error estimates in the $L^{2}$ and the $H^{1}$ norm, under certain assumptions on the partition of the time interval and avoiding the enforcement of a Courant-Friedrichs-Lewy (CFL) condition between the space mesh size and the time step sizes.

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论非线性薛定谔方程线性隐式有限元方法的收敛性

我们考虑了一个二维和三维空间非线性薛定谔方程的模型初始和狄利克边界值问题。该问题的解是通过一种保守的数值方法逼近的，该方法由标准的符合有限元空间离散化和二阶线性隐式时间步进组成，在时间间隔的非均匀分区的节点和中点处产生近似值。我们研究了该方法的收敛性，在时间间隔分区的某些假设条件下，并避免在空间网格大小和时间步长之间执行库朗-弗里德里希斯-勒维（CFL）条件，推导出了最优阶误差估计值和规范。

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来源期刊

Studies in Applied Mathematics 数学-应用数学

CiteScore

4.30

自引率

3.70%

发文量

审稿时长

>12 weeks

期刊介绍： Studies in Applied Mathematics explores the interplay between mathematics and the applied disciplines. It publishes papers that advance the understanding of physical processes, or develop new mathematical techniques applicable to physical and real-world problems. Its main themes include (but are not limited to) nonlinear phenomena, mathematical modeling, integrable systems, asymptotic analysis, inverse problems, numerical analysis, dynamical systems, scientific computing and applications to areas such as fluid mechanics, mathematical biology, and optics.