Two-Level discretization techniques for ground state computations of Bose-Einstein condensates

TL;DR

This paper introduces a two-level finite element discretization method for efficiently computing the ground states of Bose-Einstein condensates, achieving high accuracy with reduced computational effort.

Contribution

The novel two-scale discretization approach constructs a low-dimensional space for solving the nonlinear eigenvalue problem, improving efficiency without sacrificing accuracy.

Findings

Achieves convergence rates of H^3 for eigenfunctions and H^4 for eigenvalues.

Preprocessing step is independent of boson types and numbers.

Numerical experiments suggest actual convergence may surpass theoretical predictions.

Abstract

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.