Efficient finite element methods for semiclassical nonlinear Schrödinger equations with random potentials
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866929710279163904 |
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| author | Li, Panchi Zhang, Zhiwen |
| author_facet | Li, Panchi Zhang, Zhiwen |
| contents | In this paper, we propose two time-splitting finite element methods to solve the semiclassical nonlinear Schrödinger equation (NLSE) with random potentials. We then introduce the multiscale finite element method (MsFEM) to reduce the degrees of freedom in the physical space. We construct multiscale basis functions by solving optimization problems and rigorously analyze two time-splitting MsFEMs for the semiclassical NLSE with random potentials. We provide the $L^2$ error estimate of the proposed methods and show that they achieve second-order accuracy in both spatial and temporal spaces and an almost first-order convergence rate in the random space. Additionally, we present a multiscale reduced basis method to reduce the computational cost of constructing basis functions for solving random NLSEs. Finally, we carry out several 1D and 2D numerical examples to validate the convergence of our methods and investigate wave propagation behaviors in the NLSE with random potentials. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_07569 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Efficient finite element methods for semiclassical nonlinear Schrödinger equations with random potentials Li, Panchi Zhang, Zhiwen Numerical Analysis 35Q55, 65M60, 81Q05, 47H40 In this paper, we propose two time-splitting finite element methods to solve the semiclassical nonlinear Schrödinger equation (NLSE) with random potentials. We then introduce the multiscale finite element method (MsFEM) to reduce the degrees of freedom in the physical space. We construct multiscale basis functions by solving optimization problems and rigorously analyze two time-splitting MsFEMs for the semiclassical NLSE with random potentials. We provide the $L^2$ error estimate of the proposed methods and show that they achieve second-order accuracy in both spatial and temporal spaces and an almost first-order convergence rate in the random space. Additionally, we present a multiscale reduced basis method to reduce the computational cost of constructing basis functions for solving random NLSEs. Finally, we carry out several 1D and 2D numerical examples to validate the convergence of our methods and investigate wave propagation behaviors in the NLSE with random potentials. |
| title | Efficient finite element methods for semiclassical nonlinear Schrödinger equations with random potentials |
| topic | Numerical Analysis 35Q55, 65M60, 81Q05, 47H40 |
| url | https://arxiv.org/abs/2502.07569 |