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