Multigrid shape optimization governed by elliptic PDEs (Q2840143)

The authors introduce the shape optimization problem by considering the cost functional NEWLINE\[NEWLINEJ(y,\Omega):= \int_\Omega r(y)\,d\Omega+ {\lambda_1\over 2}\,\Biggl(\int_{\partial\Omega} d\Gamma- P\Biggr)^2+ {\lambda_2\over 2}\,\Biggl(\int_\Omega d\Omega- A\Biggr)^2,NEWLINE\]NEWLINE where \(y= y(\Omega)\) is the unique solution of the elliptic partial differential equation NEWLINE\[NEWLINE\begin{aligned} -\Delta y= f\quad &\text{in }\Omega,\\ y= y_b\quad &\text{on }\delta\Omega.\end{aligned}NEWLINE\]NEWLINE The authors present and analyse a new multigrid framework to solve shape problems. The convergence of the proposed multigrid shape optimization method is proved.NEWLINENEWLINENumerical experiments are presented.