Finite element approximations of an optimal control problem associated with the scalar Ginzburg-Landau equation
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.
scientific article; zbMATH DE number 4192973