A multi-mesh adaptive finite element method for solving the Gross-Pitaevskii equation

Abstract: It is found that the wave functions of the Gross-Pitaevskii equation (GPE) often vary significantly in different spatial regions, with some components exhibiting sharp variations while others remain smooth. Solving the GPE on a single mesh, even with adaptive refinement, can lead to excessive computational costs due to the need to accommodate the most oscillatory solution. To address this issue, we present a multi-mesh adaptive finite element method for solving the GPE. To this end, we first convert it into a time-dependent equation through the imaginary time propagation method. Then the equation is discretized by the backward Euler method temporally and the multi-mesh adaptive finite element method spatially. The proposed method is compared with the single-mesh adaptive method through a series of numerical experiments, which demonstrate that the multi-mesh adaptive method can achieve the same numerical accuracy with less computational consumption.