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

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 $u^{\ast }\in X\subseteq H^1(\mathrm{\Omega })$ the exact solution to such an equation, by $\{X_{\delta }{\}}_{\delta >0}$, a sequence of conforming subspaces of $X$ satisfying the approximation property, by $u_{\delta }^{\ast }\in X_{\delta }$ the Galerkin solution to the equation, and by ${\mathrm{\Pi }}_{\delta }^X{u}^{\ast }$, the $(\cdot ,\cdot {)}_X$-best approximation in $X_{\delta }$ of $u^{\ast }$, we show – under some assumptions – that $u_{\delta }^{\ast }$ converges at a higher rate to ${\mathrm{\Pi }}_{\delta }^X{u}^{\ast }$ than to $u^{\ast }$ 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.