On the relation between Galerkin approximations and canonical best-approximations of solutions to some non-linear Schrödinger equations

TL;DR

The paper analyzes Galerkin approximations to nonlinear Schrödinger/Gross–Pitaevskii–type energy minimisation problems and proves superconvergence: the Galerkin solutions converge faster to the $(\cdot,\cdot)_X$-best approximations of the exact minimisers than to the exact minimisers themselves, in both $L^2$ and $H^1$ norms. It develops an abstract framework with Assumptions $A.1$–$A.3$ for the source problem and $B.1$–$B.3$ for the constrained (eigenvalue) problem, leveraging auxiliary potentials and adjoint problems to derive improved rates. The results are then instantiated for three discretisation families: conforming finite elements, spectral polynomials, and spectral Fourier methods, yielding explicit rate enhancements (up to $N^{-3+\varepsilon}$ or $N^{-3}$) under suitable regularity assumptions on the data and solution. This work provides near-optimal, a priori error estimates that justify near-optimal iteration strategies in nonlinear periodic eigenvalue computations and quantum-chemistry-type simulations. The analysis integrates elliptic regularity, projection theory, and Aubin–Nitsche techniques in a nontrivial nonlinear setting.

Abstract

In this paper, we establish a superconvergence property of Galerkin approximations to some non-linear Schrödinger equations of Gross-Pitaevskii type. More precisely, denoting by $u^*\in X \subseteq H^1(Ω)$ the exact solution to such an equation, by $\{X_δ\}_{δ>0}$, a sequence of conforming subspaces of $X$ satisfying the approximation property, by $u_δ^*\in X_δ$ the Galerkin solution to the equation, and by $Π^X_δ u^*$, the $(\cdot, \cdot)_{X}$-best approximation in $X_δ$ of $u^*$, we show -- under some assumptions -- that $u_δ^*$ converges at a higher rate to $Π^X_δ u^*$ than to $u^*$ in both the $L^2$ norm and the canonical $H^1$ norm. Our results apply to conforming finite element discretisations as well as spectral Galerkin methods based on polynomials or Fourier (plane-wave) expansions.

Theorems & Definitions (22)

• Remark 1: A Sufficient Condition to Guarantee Assumption A.2
• Definition 2
• Lemma 3
• proof
• Lemma 4
• proof
• Lemma 5
• proof
• Theorem 6
• proof