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

TL;DR

This paper applies Dirichlet-Neumann and Neumann-Neumann methods to elliptic PDE control problems with $H^{-1}$ regularization, simplifying the solution of singular perturbed Poisson equations and analyzing their effectiveness in 1D and 2D.

Contribution

It introduces the use of DN and NN methods for $H^{-1}$ regularized elliptic control problems, avoiding coupled bi-Laplacian solutions and providing detailed analysis and numerical results.

Findings

DN and NN methods effectively solve $H^{-1}$ regularized problems.

The approach simplifies handling less regular solutions.

Numerical experiments confirm the methods' efficiency.

Abstract

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.