In this paper, we consider an energy-conserving continuous Galerkin discretization of the Gross-Pitaevskii equation with a magnetic trapping potential and a stirring potential for angular momentum rotation. The discretization is based on finite elements in space and time and allows for arbitrary polynomial orders. It was first analyzed in [O. Karakashian, C. Makridakis; SIAM J. Numer. Anal. 36(6):1779-1807, 1999] in the absence of potential terms and corresponding a priori error estimates were derived in 2D. In this work we revisit the approach in the generalized setting of the Gross-Pitaevskii equation with rotation and we prove uniform $L^\infty$-bounds for the corresponding numerical approximations in 2D and 3D without coupling conditions between the spatial mesh size and the time step size. With this result at hand, we are in particular able to extend the previous error estimates to the 3D setting while avoiding artificial CFL conditions.

翻译：本文研究带磁陷阱势和角动量旋转搅拌势的Gross-Pitaevskii方程的能量守恒连续Galerkin离散化。该离散化采用时空有限元方法，支持任意多项式阶数。该方法最初在[O. Karakashian, C. Makridakis; SIAM J. Numer. Anal. 36(6):1779-1807, 1999]中针对无势能项的情形进行分析，并推导了二维先验误差估计。本研究在含旋转项Gross-Pitaevskii方程的广义框架下重新审视该方法，证得二维与三维情形下数值近似解的均匀$L^\infty$界，且无需空间网格尺寸和时间步长之间的耦合条件。基于该结果，我们特别能够将先前的误差估计推广至三维情形，同时避免人为的CFL条件。