A Monolithic Discontinuous Galerkin Framework for Darcy Optimal Control with Radon-Measure Tracking and Pointwise Control Constraints
Abstract
We study an elliptic optimal control problem governed by Darcy's equation in heterogeneous porous media, with pointwise box constraints on the control. The objective functional is formulated via a Radon measure, which allows the desired pressure state to be tracked on observation sets of varying dimension, including points, curves, and subdomains, within a single formulation. The state and adjoint equations are discretized by a symmetric interior penalty discontinuous Galerkin method, yielding a locally mass-conservative approximation that is robust across strong permeability discontinuities, while the control is approximated by piecewise constants. The state, adjoint, and control are retained as primary unknowns in a single coupled optimality system and solved monolithically by a primal-dual active set strategy. We establish stability and well-posedness of the discretization and derive a priori L2 error estimates for both variables. The principal difficulty is the reduced regularity of the adjoint state induced by the measure-valued tracking data. The analysis controls the resulting adjoint-control coupling through an intermediate adjoint driven by the continuous optimal state. Numerical experiments confirm the predicted convergence rates for point, curve, and subdomain observations, exhibit mesh-independent primal-dual active set iteration counts, and demonstrate local mass conservation.
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BibTeX
@misc{jeong2026monolithic,
title = {{A Monolithic Discontinuous Galerkin Framework for Darcy Optimal Control with Radon-Measure Tracking and Pointwise Control Constraints}},
author = {SeongHee Jeong and Sanghyun Lee},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.30707},
eprint = {2609.30707},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.30707},
}
RIS
TY - UNPB
TI - A Monolithic Discontinuous Galerkin Framework for Darcy Optimal Control with Radon-Measure Tracking and Pointwise Control Constraints
AU - SeongHee Jeong
AU - Sanghyun Lee
PY - 2026
DA - 2026/09/25
PB - arXiv
DO - 10.48550/arxiv.2609.30707
UR - https://arxiv.org/abs/2609.30707
AB - We study an elliptic optimal control problem governed by Darcy's equation in heterogeneous porous media, with pointwise box constraints on the control. The objective functional is formulated via a Radon measure, which allows the desired pressure state to be tracked on observation sets of varying dimension, including points, curves, and subdomains, within a single formulation. The state and adjoint equations are discretized by a symmetric interior penalty discontinuous Galerkin method, yielding a locally mass-conservative approximation that is robust across strong permeability discontinuities, while the control is approximated by piecewise constants. The state, adjoint, and control are retained as primary unknowns in a single coupled optimality system and solved monolithically by a primal-dual active set strategy. We establish stability and well-posedness of the discretization and derive a priori L^2 error estimates for both variables. The principal difficulty is the reduced regularity of the adjoint state induced by the measure-valued tracking data. The analysis controls the resulting adjoint-control coupling through an intermediate adjoint driven by the continuous optimal state. Numerical experiments confirm the predicted convergence rates for point, curve, and subdomain observations, exhibit mesh-independent primal-dual active set iteration counts, and demonstrate local mass conservation.
ER -