On the relation between Galerkin approximations and canonical best-approximations of solutions to some non-linear Schr\"odinger equations

TL;DR

This paper proves that Galerkin approximations for certain non-linear Schrödinger equations converge faster to the best possible approximations within the discretization space than to the exact solution, under specific conditions.

Contribution

It establishes a superconvergence property of Galerkin methods for non-linear Schrödinger equations, linking approximation quality to best-approximations in the discretization space.

Findings

Galerkian solutions converge faster to best approximations than to the exact solution.

Results apply to finite element and spectral Galerkin methods.

Superconvergence occurs under certain assumptions.

Abstract

In this paper, we establish a superconvergence property of Galerkin approximations to some non-linear Schr\"odinger equations of Gross-Pitaevskii type. More precisely, denoting by $u^*\in X \subseteq H^1(\Omega)$ the exact solution to such an equation, by $\{X_{\delta}\}_{\delta >0}$, a sequence of conforming subspaces of $X$ satisfying the approximation property, by $u_\delta^*\in X_{\delta}$ the Galerkin solution to the equation, and by $\Pi^X_{\delta} u^*$, the $(\cdot, \cdot)_{X}$-best approximation in $X_\delta$ of $u^*$, we show -- under some assumptions -- that $u_\delta^*$ converges at a higher rate to $\Pi^X_{\delta} u^*$ 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.