Uniform L∞-bounds for energy-conserving higher-order time integrators for the Gross–Pitaevskii equation with rotation

Abstract

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 by O. Karakashian and 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 particularly able to extend the previous error estimates to the $3D$ setting while avoiding artificial CFL conditions.