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Multigrid Preconditioning for Discontinuous Galerkin Discretizations of an Elliptic Optimal Control Problem with a Convection-Dominated State Equation

We consider discontinuous Galerkin methods for an elliptic distributed optimal control problem constrained by a convection-dominated problem. We prove global optimal convergence rates using an inf-sup condition, with the diffusion parameter ε and regularization parameter β explicitly tracked. We the...

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Published in:	Journal of scientific computing 2024-12, Vol.101 (3), p.79, Article 79
Main Authors:	Liu, Sijing, Simoncini, Valeria
Format:	Article
Language:	English
Subjects:	Algorithms Computational Mathematics and Numerical Analysis Convection Diffusion barriers Diffusion rate Discretization Equations of state Galerkin method Mathematical and Computational Engineering Mathematical and Computational Physics Mathematics Mathematics and Statistics Multigrid methods Numerical analysis Optimal control Parameters Preconditioning Regularization Theoretical
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description	We consider discontinuous Galerkin methods for an elliptic distributed optimal control problem constrained by a convection-dominated problem. We prove global optimal convergence rates using an inf-sup condition, with the diffusion parameter ε and regularization parameter β explicitly tracked. We then propose a multilevel preconditioner based on downwind ordering to solve the discretized system. The preconditioner only requires two approximate solves of single convection-dominated equations using multigrid methods. Moreover, for the strongly convection-dominated case, only two sweeps of block Gauss-Seidel iterations are needed. We also derive a simple bound indicating the role played by the multigrid preconditioner. Numerical results are shown to support our findings.
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subjects	Algorithms Computational Mathematics and Numerical Analysis Convection Diffusion barriers Diffusion rate Discretization Equations of state Galerkin method Mathematical and Computational Engineering Mathematical and Computational Physics Mathematics Mathematics and Statistics Multigrid methods Numerical analysis Optimal control Parameters Preconditioning Regularization Theoretical
title	Multigrid Preconditioning for Discontinuous Galerkin Discretizations of an Elliptic Optimal Control Problem with a Convection-Dominated State Equation
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