Multigrid preconditioning for discontinuous Galerkin discretizations of an elliptic optimal control problem with a convection-dominated state equation

2024/06/13 by Sijing Liu, Valeria Simoncini, Liu, Sijing +1 · 1 citation
Engineering · Computer Science · #Advanced Numerical Methods in Computational Mathematics #Advanced Mathematical Modeling in Engineering #Computational Fluid Dynamics and Aerodynamics

paper · pdf · doi:10.48550/arxiv.2406.09276

Abstract

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.

Cited by

• Convergence analysis of a balancing domain decomposition method for an elliptic optimal control problem with HDG discretizations

Related

• Robust Multigrid Methods for Discontinuous Galerkin Discretizations of an Elliptic Optimal Control Problem
• A balancing domain decomposition by constraints preconditioner for a hybridizable discontinuous Galerkin discretization of an elliptic optimal control problem
• A fast and memory-efficient spectral Galerkin scheme for distributed elliptic optimal control problems
• Space-Time Discontinuous Galerkin Solution of Convection Dominated Optimal Control Problems
• Multigrid solution of a distributed optimal control problem constrained by the Stokes equations