Analysis of Dirichlet-Neumann and Neumann-Dirichlet Methods for Time-Periodic Parabolic Optimal Control Problems

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Dirichlet-Neumann and Neumann-Neumann methods are not only the parallel strategies in the spatial domain, but also can be used as a class of parallel methods in time. In this paper, we propose the Dirichlet-Neumann and Neumann-Dirichlet algorithms for time-periodic parabolic optimal control problems. By the Lagrange multiplier approach, a coupled system is obtained with a special time-periodic condition. For this coupled system, the Dirichlet-Neumann and Neumann-Dirichlet algorithms and their three variants are derived. We present the convergence analysis for all proposed algorithms. The numerical performance of the convergence factors is shown to illustrate our theoretical analysis. Based on our analysis, there is a class of algorithms with the better convergence compared with the natural Dirichlet-Neumann algorithm. Finally, numerical experiments are provided to illustrate the theoretical results.

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