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Preprints, Working Papers, ... Year : 2013

A Posteriori Analysis of a Non-Linear Gross-Pitaevskii type Eigenvalue Problem

Geneviève Dusson
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Yvon Maday

Abstract

In this paper, we provide a first full {\it a posteriori} error analysis for variational approximations of the ground state eigenvector of a non-linear elliptic problems of the Gross-Pitaevskii type, more precisely of the form $-\Delta u + Vu + u^3 = \lambda u$, $\|u\|_{L^2}=1$, with periodic boundary conditions in one dimension. Denoting by $(u_N,\lambda_N)$ the variational approximation of the ground state eigenpair $(u,\lambda)$ based on a Fourier spectral approximation and $(u_N^k,\lambda_N^k)$ the approximate solution at the $k^{th}$ iteration of an algorithm used to solve the non-linear problem, we first provide a precised \textit{a priori} analysis of the convergence rates of $\|u-u_N\|_{H^1}$, $\|u-u_N\|_{L^2}$, $|\lambda-\lambda_N|$ and then present original \textit{a posteriori} estimates in the convergence rates of $\|u_-u_N^k\|_{H^1}$ when $N$ and $k$ go to infinity. We introduce a residue standing for the global error $R_N^k=-\Delta u_N^k+Vu_N^k+(u_N^k)^3-\lambda_N^ku_N^k$ and we divide it into two residues characterizing respectively the error due to the discretization of the space and the finite number of iterations when solving the problem numerically. We show that the numerical results are coherent with this \textit{a posteriori} analysis.
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Dates and versions

hal-00903715 , version 1 (12-11-2013)

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  • HAL Id : hal-00903715 , version 1

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Geneviève Dusson, Yvon Maday. A Posteriori Analysis of a Non-Linear Gross-Pitaevskii type Eigenvalue Problem. 2013. ⟨hal-00903715⟩
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