Numerical vortex resolution for the Gross-Pitaevskii equation in the rapid rotation Thomas-Fermi scaling

In this paper we analyze finite element approximations of ground states of the Gross-Pitaevskii equation in the rapid rotation Thomas-Fermi scaling.

In this regime, the healing length and vortex core size are of order $\eps \ll 1$, while the effective confinement potential may degenerate as the angular velocity approaches a critical value.

In this setting, we analyze the $\eps$-dependence of the ground states and show that the local flatness of the energy landscape plays a decisive role for numerical resolution.

More precisely, we establish mesh size conditions that guarantee the existence of discrete ground states in finite element spaces which are quasi-best approximations of an exact ground state.

In particular, we prove that the absolute $H^1$-error behaves asymptotically like $h/\eps^2$.

However, to enter this asymptotic regime, the mesh size must satisfy a significantly stronger resolution condition than the natural requirement $h \lesssim \eps$.

The additional restriction is governed by the first spectral gap of the Riemannian Hessian of the energy functional at the ground state, which measures the local flatness of the energy surface.

With this, our results provide an explanation of the mesh resolution required to capture vortex structures in rapidly rotating Bose-Einstein condensates and highlight the interplay between vortex core size, spectral stability, and discretization accuracy.