Optimized Schwarz Methods For The Optimal Control Of Systems Governed By Elliptic Partial Differential Equations

Optimal control of systems governed by elliptic partial differential equations(PDEs)without constraints can be equivalently reformulated as a coupled system of second order elliptic PDEs,which has been considered to solve by a Schwarz domain decomposition method with Robin like or coupled transmission conditions by Benamou [J.D.BENAMOU,SIAM J.Numer.Anal.,33(1996),pp.2401–2416.],where convergence analysis had been performed.Towards fast convergence of the Schwarz subdomain iterations,in this paper we firstly perform,for fixed Tikhonov parameter ,rigorous analyses based on optimization of the convergence factor of subdomain iterations in Fourier frequency domain to give the optimized transmission parameters involved in the above mentioned transmission conditions,as well as those involved in a Ventcell like and a two-sided Robin like transmission conditions that we propose to accelerate the Schwarz subdomain iterations,and meanwhile we obtain also the corresponding asymptotic convergence rate estimates.The results show that the Tikhonov parameter occurs in both the optimized transmission parameters and the corresponding convergence rate estimates and affects the performance of the Schwarz domain decomposition significantly: with the decreasing of the Tikhonov parameter ,the subdomain iterations becomes more and more fast,though the regularity of the system deteriorates in this process.We lastly investigate the case where the Tikhonov parameter = ?4that is suggested by Benamou(where ? is the mesh size).We obtain as well the optimized transmission parameters involved in the Robin like and the coupled transmission conditions,and we find that they lead to almost constant convergence rates.The analysis also shed light on optimizing the Schwarz domain decomposition methods for biharmonic equations,since they can also be reformulated as a system of elliptic PDEs.We use various numerical experiments to validate the theoretical findings.