Anna Persson
Division of Scientific Computing, IT Department, Uppsala University
Bose-Einstein condensates are formed when a gas of Bosons are cooled to ultra-low temperatures. At this point, most of the Bosons occupy the same quantum state and behave like one giant ``macro particle''. This is classified as a new state of matter, where some quantum mechanical phenomena can be observed macroscopically. One such phenomenon is superfluidity, which describes a flow without inner friction.
To compute ground states of Bose-Einstein condensates the stationary Gross-Pitaevskii equation (GPE) is often used. In this talk we revisit a two-level multiscale technique introduced in (P.Henning, A.Målqvist, D. Petersim '14) for computing numerical approximations to the GPE. The method is based on localized orthogonal decomposition of the solution space, which exhibit high approximation properties and improves the convergence rates compared to classical finite element methods. This reduces the computational cost for computing the ground states significantly.
In the previous paper, high order convergence of the method was proven, but even higher orders were observed numerically. In this talk we show how to improve the analysis to achieve the observed rates, which are as high as O(H^6) for the eigenvalues. We also show some numerical experiments for both smooth and discontinuous potentials.