Dirichlet-Neumann and Neumann-Neumann Methods for Elliptic Control Problems

Autor:	Gander, Martin Jakob, Lu, Liu-Di
Rok vydání:	2023
Předmět:	Mathematics - Numerical Analysis Mathematics - Optimization and Control
Druh dokumentu:	Working Paper
Popis:	We present the Dirichlet-Neumann (DN) and Neumann-Neumann (NN) methods applied to the optimal control problems arising from elliptic partial differential equations (PDEs) under the $H^{-1}$ regularization. We use the Lagrange multiplier approach to derive a forward-backward optimality system with the $L^2$ regularization, and a singular perturbed Poisson equation with the $H^{-1}$ regularization. The $H^{-1}$ regularization thus avoids solving a coupled bi-Laplacian problem, yet the solutions are less regular. The singular perturbed Poisson equation is then solved by using the DN and NN methods, and a detailed analysis is given both in the one-dimensional and two-dimensional case. Finally, we provide some numerical experiments with conclusions. Comment: 27th International Domain Decomposition Conference (DD27)
Databáze:	arXiv
Externí odkaz:	http://arxiv.org/abs/2308.12764 Zobrazit plný text záznamu View this record from Arxiv