7 research outputs found
The Finite Element Method for the time-dependent Gross-Pitaevskii equation with angular momentum rotation
We consider the time-dependent Gross-Pitaevskii equation describing the
dynamics of rotating Bose-Einstein condensates and its discretization with the
finite element method. We analyze a mass conserving Crank-Nicolson-type
discretization and prove corresponding a priori error estimates with respect to
the maximum norm in time and the - and energy-norm in space. The estimates
show that we obtain optimal convergence rates under the assumption of
additional regularity for the solution to the Gross-Pitaevskii equation. We
demonstrate the performance of the method in numerical experiments
On discrete ground states of rotating Bose-Einstein condensates
The ground states of Bose-Einstein condensates in a rotating frame can be
described as constrained minimizers of the Gross-Pitaevskii energy functional
with an angular momentum term. In this paper we consider the corresponding
discrete minimization problem in Lagrange finite element spaces of arbitrary
polynomial order and we investigate the approximation properties of discrete
ground states. In particular, we prove a priori error estimates of optimal
order in the - and -norm, as well as for the ground state energy and
the corresponding chemical potential. A central issue in the analysis of the
problem is the missing uniqueness of ground states, which is mainly caused by
the invariance of the energy functional under complex phase shifts. Our error
analysis is therefore based on an Euler-Lagrange functional that we restrict to
certain tangent spaces in which we have local uniqueness of ground states. This
gives rise to an error decomposition that is ultimately used to derive the
desired a priori error estimates. We also present numerical experiments to
illustrate various aspects of the problem structure
Uniform -bounds for energy-conserving higher-order time integrators for the Gross-Pitaevskii equation with rotation
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 -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
A two level approach for simulating Bose-Einstein condensates by localized orthogonal decomposition
In this work, we consider the numerical computation of ground states and
dynamics of single-component Bose-Einstein condensates (BECs). The
corresponding models are spatially discretized with a multiscale finite element
approach known as Localized Orthogonal Decomposition (LOD). Despite the
outstanding approximation properties of such a discretization in the context of
BECs, taking full advantage of it without creating severe computational
bottlenecks can be tricky. In this paper, we therefore present two
fully-discrete numerical approaches that are formulated in such a way that they
take special account of the structure of the LOD spaces. One approach is
devoted to the computation of ground states and another one for the computation
of dynamics. A central focus of this paper is also the discussion of
implementation aspects that are very important for the practical realization of
the methods. In particular, we discuss the use of suitable data structures that
keep the memory costs economical. The paper concludes with various numerical
experiments in 1d, 2d and 3d that investigate convergence rates and
approximation properties of the methods and which demonstrate their performance
and computational efficiency, also in comparison to spectral and standard
finite element approaches