Finite element approximations of an optimal control problem associated with the scalar Ginzburg-Landau equation
type: Item
instance of: scholarly article
title: Finite element approximations of an optimal control problem associated with the scalar Ginzburg-Landau equation
zbMATH Open document ID: 0723.65043
DOI: 10.1016/0898-1221(91)90090-Q
author: Max Gunzburger
author: L. Hou
author: Thomas P. Svobodny
publication date: 1991-01-01T00:00:00Z
review text: The existence of an optimal solution for the problem \((1)\quad 1/2\int_{\Omega}(u-u_ 0)^ 2d\Omega +1/2\int_{\Gamma}g^ 2d\Gamma \to \min_{(u,g)},\) \(-\Delta u+u^ 3-u=0,\quad u\in H^ 1(\Omega);\quad \frac{\partial u}{\partial n}|_{\Gamma}=g,\quad g\in L^ 2(\Gamma);\quad u_ 0\in L^ 2(\Omega)\) is proved. The use of Lagrange multipliers for (1) is justified and an optimality system of equations is derived. The \(H^ 2(\Omega)\)-regularity of the optimal solution \(\hat u\) and the associated Lagrange multiplier is stated. A finite element algorithm for solving the optimality system is defined and optimal error estimates are obtained.
Mathematics Subject Classification ID: 65K10
Mathematics Subject Classification ID: 49M15
Mathematics Subject Classification ID: 49K20
zbMATH DE Number: 4192973
zbMATH Keywords: optimal control problem
zbMATH Keywords: scalar Ginzburg-Landau equation
zbMATH Keywords: optimal solution
zbMATH Keywords: Lagrange multipliers
zbMATH Keywords: finite element algorithm
zbMATH Keywords: optimal error estimates
reviewed by: Mikhail Yu. Kokurin
MaRDI profile type: Publication
cites work: Q4073917
cites work: Finite dimensional approximation of nonlinear problems. I: Branches of nonsingular solutions
cites work: Q4162607
cites work: Finite element approximation of the Navier-Stokes equations
published in: Computers & Mathematics with Applications
description: scientific article; zbMATH DE number 4192973