Parallel finite-element codes for the Bogoliubov-de Gennes stability analysis of Bose-Einstein condensates

We present and distribute a parallel finite-element toolbox written in the free software FreeFEM for computing the Bogoliubov-de Gennes (BdG) spectrum of stationary solutions to one- and two-component Gross-Pitaevskii (GP) equations, in two or three spatial dimensions. The parallelization of the toolbox relies exclusively upon the recent interfacing of FreeFEM with the PETSc library. The latter contains itself a wide palette of state-of-the-art linear algebra libraries, graph partitioners, mesh generation and domain decomposition tools, as well as a suite of eigenvalue solvers that are embodied in the SLEPc library. Within the present toolbox, stationary states of the GP equations are computed by a Newton method. Branches of solutions are constructed using an adaptive step-size continuation algorithm. The combination of mesh adaptivity tools from FreeFEM with the parallelization features from PETSc makes the toolbox efficient and reliable for the computation of stationary states. Their BdG spectrum is computed using the SLEPc eigenvalue solver. We perform extensive tests and validate our programs by comparing the toolbox's results with known theoretical and numerical findings that have been reported in the literature.

我们提出并发布了一款基于自由软件FreeFEM编写的并行有限元工具箱，用于计算二维或三维空间中单组分与双组分Gross-Pitaevskii（GP）方程定态解的Bogoliubov-de Gennes（BdG）能谱。该工具箱的并行化完全依托FreeFEM与PETSc库的最新接口实现。PETSc库本身集成了丰富的尖端线性代数库、图划分器、网格生成与区域分解工具，以及内嵌于SLEPc库的一系列本征值求解器。在本工具箱中，GP方程的定态通过牛顿法求解；解的分支则通过自适应步长延续算法构建。结合FreeFEM的网格自适应工具与PETSc的并行化特性，该工具箱在定态求解任务中兼具高效性与可靠性。其BdG能谱通过SLEPc本征值求解器计算得到。我们开展了大量测试，并通过将工具箱的计算结果与文献中已报道的已知理论与数值结论进行对比，验证了程序的正确性。