Localized Orthogonal Decomposition Methods vs. Classical FEM for the Gross-Pitaevskii Equation
Christian Döding
TL;DR
The paper studies numerical solutions of the time-dependent GPE $i\partial_t u=-\Delta u+V u+\beta|u|^2u$ on bounded domains, assessing a multiscale Localized Orthogonal Decomposition (LOD) spatial discretization paired with an energy-preserving continuous Galerkin time integrator. It demonstrates that LOD attains third-order spatial accuracy and remains robust for rough potentials, outperforming classical high-order FEM in low-regularity settings while remaining competitive for smooth problems. The approach leverages a projected density in the nonlinear term, localized correctors with exponential decay, and a modified energy $E_{LOD}$ to ensure stability, offering a practical solver for Bose-Einstein condensate dynamics and multiscale GPE problems. These results highlight the method’s potential for efficient, accurate simulations where standard FEM struggles with irregular coefficients.
Abstract
The time-dependent Gross-Pitaevksii equation (GPE) is a nonlinear Schrödinger equation which is used in quantum physics to model the dynamics of Bose-Einstein condensates. In this work we consider numerical approximations of the GPE based on a multiscale approach known as the localized orthogonal decomposition. Combined with an energy preserving time integrator one derives a method which is of high order in space and time under mild regularity assumptions. In previous work, the method has been shown to be numerically very efficient compared to first order Lagrange FEM. In this paper, we further investigate the performance of the method and compare it with higher order Lagrange FEM. For rough problems we observe that the novel method performs very efficient and retains its high order, while the classical methods can only compete well for smooth problems.