49 research outputs found
Two-Level discretization techniques for ground state computations of Bose-Einstein condensates
This work presents a new methodology for computing ground states of
Bose-Einstein condensates based on finite element discretizations on two
different scales of numerical resolution. In a pre-processing step, a
low-dimensional (coarse) generalized finite element space is constructed. It is
based on a local orthogonal decomposition and exhibits high approximation
properties. The non-linear eigenvalue problem that characterizes the ground
state is solved by some suitable iterative solver exclusively in this
low-dimensional space, without loss of accuracy when compared with the solution
of the full fine scale problem. The pre-processing step is independent of the
types and numbers of bosons. A post-processing step further improves the
accuracy of the method. We present rigorous a priori error estimates that
predict convergence rates H^3 for the ground state eigenfunction and H^4 for
the corresponding eigenvalue without pre-asymptotic effects; H being the coarse
scale discretization parameter. Numerical experiments indicate that these high
rates may still be pessimistic.Comment: Accepted for publication in SIAM J. Numer. Anal., 201
Sobolev gradient flow for the Gross-Pitaevskii eigenvalue problem: global convergence and computational efficiency
We propose a new normalized Sobolev gradient flow for the Gross-Pitaevskii
eigenvalue problem based on an energy inner product that depends on time
through the density of the flow itself. The gradient flow is well-defined and
converges to an eigenfunction. For ground states we can quantify the
convergence speed as exponentially fast where the rate depends on spectral gaps
of a linearized operator. The forward Euler time discretization of the flow
yields a numerical method which generalizes the inverse iteration for the
nonlinear eigenvalue problem. For sufficiently small time steps, the method
reduces the energy in every step and converges globally in to an
eigenfunction. In particular, for any nonnegative starting value, the ground
state is obtained. A series of numerical experiments demonstrates the
computational efficiency of the method and its competitiveness with established
discretizations arising from other gradient flows for this problem
A numerical study of vortex nucleation in 2D rotating Bose-Einstein condensates
This article introduces a new numerical method for the minimization under
constraints of a discrete energy modeling multicomponents rotating
Bose-Einstein condensates in the regime of strong confinement and with
rotation. Moreover, we consider both segregation and coexistence regimes
between the components. The method includes a discretization of a continuous
energy in space dimension 2 and a gradient algorithm with adaptive time step
and projection for the minimization. It is well known that, depending on the
regime, the minimizers may display different structures, sometimes with
vorticity (from singly quantized vortices, to vortex sheets and giant holes).
In order to study numerically the structures of the minimizers, we introduce in
this paper a numerical algorithm for the computation of the indices of the
vortices, as well as an algorithm for the computation of the indices of vortex
sheets. Several computations are carried out, to illustrate the efficiency of
the method, to cover different physical cases, to validate recent theoretical
results as well as to support conjectures. Moreover, we compare this method
with an alternative method from the literature