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arXiv · 2309.05967

Optimal $L^2$ error estimates of mass- and energy-conserved FE schemes for a nonlinear Schr\"odinger-type system

Zhuoyue Zhang·Wentao Cai
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Abstract

In this paper, we present an implicit Crank-Nicolson finite element (FE) scheme for solving a nonlinear Schr\"odinger-type system, which includes Schr\"odinger-Helmholz system and Schr\"odinger-Poisson system. In our numerical scheme, we employ an implicit Crank-Nicolson method for time discretization and a conforming FE method for spatial discretization. The proposed method is proved to be well-posedness and ensures mass and energy conservation at the discrete level. Furthermore, we prove optimal $L^2$ error estimates for the fully discrete solutions. Finally, some numerical examples are provided to verify the convergence rate and conservation properties.

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Zhuoyue Zhang, Wentao Cai. 2023-09-12. Optimal $L^2$ error estimates of mass- and energy-conserved FE schemes for a nonlinear Schr\"odinger-type system. https://arxiv.org/abs/2309.05967

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