Two-level discretization techniques for ground state computations of Bose-Einstein condensates
Journal article, 2014

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 preprocessing step, a low-dimensional (coarse) generalized finite element space is constructed. It is based on a local orthogonal decomposition of the solution space and exhibits high approximation properties. The nonlinear eigenvalue problem that characterizes the ground state is solved by some suitable iterative solver exclusively in this low-dimensional space, without significant loss of accuracy when compared with the solution of the full fine scale problem. The preprocessing step is independent of the types and numbers of bosons. A postprocessing 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.

Gross-Pitaevskii equation

multiscale method

finite element

numerical upscaling

two-grid method



P. Henning

Swiss Federal Institute of Technology in Lausanne (EPFL)

Axel Målqvist

Chalmers, Mathematical Sciences, Mathematics

University of Gothenburg

Daniel Peterseim

University of Bonn

SIAM Journal on Numerical Analysis

0036-1429 (ISSN) 1095-7170 (eISSN)

Vol. 52 4 1525-1550

Subject Categories

Computational Mathematics



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