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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Bibliographic Details
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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Summary: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.
ISSN:0885-7474
1573-7691
DOI:10.1007/s10915-024-02717-9