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

30 pages