Patrick Henning (Ruhr-University Bochum): Finite Element Resolution of Nonlinear Ground-State Problems with Small Scales

Many nonlinear variational problems involve small spatial scales that have to be resolved accurately by a finite element discretization. A natural expectation is that the mesh size should be comparable to the smallest relevant physical length scale. We show that this intuition can fail when the underlying energy landscape becomes locally flat near a minimizer.
We study this phenomenon for a class of nonlinear ground-state problems arising in the rapid-rotation regime of the Gross–Pitaevskii equation. The small parameter $\varepsilon$ determines the size of vortex cores, suggesting the natural resolution condition $h\lesssim\varepsilon$. However, as the system approaches a critical regime, the stability of the ground state deteriorates and an additional, significantly stronger mesh constraint emerges.
Our analysis relates the finite element approximation error to the local spectral stability of the continuous minimizer. In particular, we identify the first spectral gap of the Riemannian Hessian as a key quantity governing the required mesh resolution and prove conditions under which discrete minimizers are quasi-best approximations of exact ground states. We also derive the asymptotic $H^1$-error estimate and discuss how the resulting theory explains the mesh resolution required to accurately capture vortex structures.