Robust and efficient finite element multigrid and preconditioned minimum residual solvers for the distributed elliptic optimal control problems

Volume 22, Issue 2, pp 158--173 http://dx.doi.org/10.22436/jmcs.022.02.07

Publication Date: July 23, 2020 Submission Date: February 14, 2020 Revision Date: March 28, 2020 Accteptance Date: April 27, 2020

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Authors

Kizito Muzhinji - Department of Mathematics and Applied Mathematics, University of Venda, P. B. X5050, Thohoyandou 0950, South Africa.

Abstract

In this study, the optimal control problem is considered. This is an important class of partial differential equations constrained optimization problems. The constraint here is an elliptic partial differential equation with Neumann boundary conditions. The discretization of the optimality system produces a block coupled algebraic system of equations of saddle point form. The solution of such systems is a major computational task since they require specialized methods. Constructing robust, fast and efficient solvers for their numerical solution has preoccupied the computational science community for decades and various approaches have been developed. The approaches involve solving simultaneously for all the unknowns using a coupled block system, the segregated approach where a reduced system is solved and the approach of reducing to a fixed point form. Here the minimum residual solver with ideal preconditioning is applied to the unreduced 3 by 3 and reduced 2 by 2 coupled systems and compared to the multigrid method applied to the compact fixed-point form. The two methods are compared numerically in terms of iterative counts and computational times. The numerical results indicate that the two methods produce similar outcomes and the multigrid solver becoming very competitive in terms of the iterative counts though slower than preconditioned minimum residual solver in terms of computational times. For all the approaches, the two methods exhibited mesh and parameter independent convergence. The optimal performance of the two methods is verified computationally and theoretically.

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Keywords

• Elliptic optimal control problems
• partial differential equations (PDEs)
• saddle point problems
• optimality system
• finite element method (FEM)
• multigrid method (MGM)
• preconditioned minimum residual method (PMINRES)

MSC

•  49J20
•  65M22
•  65M55
•  65M60

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