Uniform $L^\infty$-bounds for energy-conserving higher-order time integrators for the Gross-Pitaevskii equation with rotation

TL;DR

This paper establishes uniform $L^$ bounds for energy-conserving Galerkin methods applied to the rotating Gross-Pitaevskii equation, enabling error analysis in 2D and 3D without restrictive mesh conditions.

Contribution

It extends previous error estimates to 3D for energy-conserving discretizations of the Gross-Pitaevskii equation with rotation, removing coupling constraints between mesh size and time step.

Findings

Proved uniform $L^$ bounds in 2D and 3D.

Extended error estimates to 3D without CFL conditions.

Validated the approach for arbitrary polynomial orders.

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 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.