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

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.