| Autor: |
Chen, Ziang, Lu, Jianfeng, Lu, Yulong, Zhang, Xiangxiong |
| Rok vydání: |
2024 |
| Předmět: |
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| Druh dokumentu: |
Working Paper |
| Popis: |
This paper studies the numerical approximation of the ground state of the Gross-Pitaevskii (GP) eigenvalue problem with a fully discretized Sobolev gradient flow induced by the $H^1$ norm. For the spatial discretization, we consider the finite element method with quadrature using $P^k$ basis on a simplicial mesh and $Q^k$ basis on a rectangular mesh. We prove the global convergence to a critical point of the discrete GP energy, and establish a local exponential convergence to the ground state under the assumption that the linearized discrete Schr\"odinger operator has a positive spectral gap. We also show that for the $P^1$ finite element discretization with quadrature on an unstructured shape regular simplicial mesh, the eigengap satisfies a mesh-independent lower bound, which implies a mesh-independent local convergence rate for the proposed discrete gradient flow. Numerical experiments with discretization by high order $Q^k$ spectral element methods in two and three dimensions are provided to validate the efficiency of the proposed method. |
| Databáze: |
arXiv |
| Externí odkaz: |
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