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

TL;DR

The paper tackles the high computational cost of solving the Gross-Pitaevskii equation (GPE) when wave functions exhibit spatially varying regularity. It introduces a multi-mesh adaptive finite element method, where each component or state is solved on its own adaptively refined mesh, and uses imaginary time propagation to evolve toward the ground state. The approach employs a hierarchical geometry tree for robust local refinement and a quadrature-based strategy for inter-mesh communication, together with a jump-type a posteriori error estimator to guide refinement. Numerical experiments show that the multi-mesh method achieves comparable numerical accuracy to single-mesh schemes while reducing degrees of freedom by substantial factors, demonstrating improved efficiency for both single-component and multi-component GPEs and extending naturally to coupled systems.

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.

Figures (9)

• Figure 1: The refinement of a triangle $\mathcal{T}_0$ and its sub-triangle $\mathcal{T}_{0,3}$.
• Figure 2: The quadtree data structure for the mesh in the right of Figure \ref{['fig:hgt']}.
• Figure 3: Twin-triangle.
• Figure 4: Hierarchical geometry tree: $\mathcal{T}_1$(left),$\mathcal{T}_2$(right).
• Figure 5: Flowchart of the multi-mesh adaptive algorithm for the coupled Gross-Pitaevskii system. $E_{old}$ is the previous energy, $tol_1$ is the difference between two adjacent solutions and $tol_2$ is adaptive tolerance; $\eta_1$, $\eta_2$ are the error indicators for mesh adaptation of $\mathcal{T}_1$ and $\mathcal{T}_2$, respectively.