一种用于快速旋转玻色-爱因斯坦凝聚体中涡旋态计算的网格自适应有限元方法
A finite element method with mesh adaptivity for computing vortex states in fast-rotating Bose-Einstein condensates
AI总结：
提出一种基于度量控制的低阶有限元网格自适应方法，结合虚时传播和Sobolev梯度下降算法，高效计算快速旋转玻色-爱因斯坦凝聚体中的涡旋态，相比均匀网格显著节省计算时间。
摘要：
快速旋转玻色-爱因斯坦凝聚体稳态的数值计算由于存在大量量子化涡旋而需要高空间分辨率。本文提出一种通过度量控制实现网格自适应的低阶有限元方法，作为常用高阶（有限差分或谱）近似方法的替代方案。网格自适应与两种不同的数值算法（虚时传播法和Sobolev梯度下降法）结合使用，用于计算稳态涡旋态。我们首先解决了用于计算网格自适应新度量的变量选择基本问题，并表明同时使用解的实部和虚部进行细化是成功的。仅使用解的模作为自适应变量的网格细化在复杂测试案例中失败。然后，我们提出了一种在解向平衡态演化过程中优化网格自适应的算法。与均匀网格计算相比，获得了可观的计算时间节省。新方法被应用于计算与物理实验相关的困难案例（大的非线性相互作用常数和高的旋转速率）。
英文摘要：
Numerical computations of stationary states of fast-rotating Bose-Einstein condensates require high spatial resolution due to the presence of a large number of quantized vortices. In this paper we propose a low-order finite element method with mesh adaptivity by metric control, as an alternative approach to the commonly used high order (finite difference or spectral) approximation methods. The mesh adaptivity is used with two different numerical algorithms to compute stationary vortex states: an imaginary time propagation method and a Sobolev gradient descent method. We first address the basic issue of the choice of the variable used to compute new metrics for the mesh adaptivity and show that simultaneously refinement using the real and imaginary part of the solution is successful. Mesh refinement using only the modulus of the solution as adaptivity variable fails for complicated test cases. Then we suggest an optimized algorithm for adapting the mesh during the evolution of the solution towards the equilibrium state. Considerable computational time saving is obtained compared to uniform mesh computations. The new method is applied to compute difficult cases relevant for physical experiments (large nonlinear interaction constant and high rotation rates).
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