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A Monolithic Discontinuous Galerkin Framework for Darcy Optimal Control with Radon-Measure Tracking and Pointwise Control Constraints

SeongHee Jeong, Sanghyun Lee

arXiv:2609.30707v1math.NAcs.NAmath.OC

TL;DR

Localized pressure targets in Darcy flow motivate a control formulation that accommodates observations on points, curves, and subdomains. The paper combines Radon-measure tracking with a locally conservative SIPG discretization and monolithic primal-dual solution; experiments confirm predicted convergence rates and local mass conservation.

  • Problem

    Darcy optimal control must steer pressure toward targets that are often available only on localized sets, while the discretization should preserve local mass balance in heterogeneous porous media.

    Source: §1 · p. 1

  • Method

    The paper uses Radon-measure tracking and coefficient-weighted SIPG with piecewise-constant control, retaining state, adjoint, and control in a monolithic primal-dual active set system.

    Source: §1 · pp. 1–2, §5 · p. 22

  • Results

    Experiments confirm predicted convergence rates and local mass conservation, while the state error suggests the sharper rate O(h2α) rather than the established O(hmin{θ,1−ϵ}) bound.

    Source: §5 · p. 22

  • Takeaways & Limitations

    The unified formulation covers pressure targets at monitoring wells, along fractures, and in selected subregions while retaining local mass conservation.

    Source: §1 · p. 3, §5 · p. 22

    The a priori error theory assumes smooth permeability K; robustness for discontinuous K is investigated numerically outside those assumptions.

    Source: §1 · p. 2

Abstract

from arXiv · show

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.

1. Introduction

The paper formulates Darcy optimal control around pressure targets on localized observation sets and develops a locally conservative DG treatment with pointwise control constraints. It combines measure-valued tracking, heterogeneous-flow discretization, and a monolithic solution of the coupled optimality system.

1. Introduction

  • Localized pressure targets reflect applications where observations or design goals occur at monitoring wells, along fractures, or in selected regions rather than throughout the domain.For Darcy flow, the discretization must also preserve local mass balance.Source: §1 · pp. 1, 3
  • The unified formulation tracks pressure at points, along curves, or in subdomains while enforcing pointwise box constraints on the control.The SIPG discretization provides local mass conservation and is designed for heterogeneous porous media.Source: §1 · pp. 1–2
  • The a priori error theory assumes smooth permeability, while the coefficient-weighted SIPG discretization is defined for discontinuous permeability and its robustness there is investigated numerically.The smoothness assumption supplies the elliptic regularity used by the error analysis.Source: §1 · p. 2
  • The state, adjoint, and control remain coupled primary unknowns in a monolithic primal-dual active set solution, preserving their discrete coupling.The control is approximated by piecewise constants rather than eliminated or reconstructed in post-processing.Source: §1 · pp. 2–3
  • Prior measure-tracking control analyses address Poisson settings and point observations, not the combination of Darcy flow, local mass conservation, and point, curve, or subdomain tracking.The present formulation also imposes constraints directly on the control.Source: §1 · p. 3

2. Well-posedness and Regularity Results

This section establishes existence and uniqueness for the measure-tracking control problem and its discrete counterpart, deriving the measure-data optimality system and the DG conditions needed for well-posedness. It also records regularity and discretization properties for error analysis, including local source balance by the numerical flux.

2.1. Radon Measure.

  • Radon-measure tracking accommodates observations on points, curves, and subdomains alongside pointwise control constraints and a locally conservative DG discretization.Overlapping observation sets are allowed; desired-state values on intersections follow the priority order point, curve, then subdomain.Source: §2 · p. 4, §2.1 · p. 4

2.2. First-Order Optimality Conditions.

  • Strict convexity and coercivity of the reduced functional yield a unique optimal control and state, while transposition defines the adjoint for its measure-valued source.The adjoint belongs to L2(Ω) and has W 1,s regularity for s < 2; the restriction reflects possible point masses, the most singular observation contribution.Source: §2.2 · pp. 5–6

2.3. Regularities.

  • The projection formula gives the optimal control W 1,s regularity for every s < 2, and satisfaction of the variational inequality characterizes the unique optimizer.The control regularity follows from the Lipschitz continuity of pointwise projection applied to the box bounds and adjoint.Source: §2.3 · p. 7

2.4. The Lagrange Multiplier.

  • The box-constraint multiplier λ = βγ̄ + z̄ and its positive and negative parts belong to W 1,s(Ω) for every s < 2.The multiplier also satisfies pointwise complementarity with the lower and upper control bounds.Source: §2.4 · p. 7

2.5. Discontinuous Galerkin Finite Element Methods.

  • The SIPG discretization uses coefficient-weighted flux averages, and a sufficiently large penalty parameter makes its bilinear form coercive in the DG energy norm.The homogeneous Dirichlet condition is imposed weakly through boundary terms.Source: §2.5 · p. 9

2.6. Discrete Problem.

  • Mesh compatibility makes the measure pairing with discontinuous finite-element functions well-defined, while projected elementwise control bounds define the discrete admissible set.The discrete bounds preserve their ordering under the stated positivity property of the elementwise L2 projection; reduced adjoint regularity affects control-error analysis.Source: §2.6 · p. 9
  • The discrete problem has a unique solution because its admissible control set is nonempty, closed, and convex in a finite-dimensional space, with β > 0.Source: §2.6 · p. 10
  • The DG well-posedness argument verifies continuity and coercivity of the bilinear form, assuming the penalty parameter σ is sufficiently large.Source: §2.6 · pp. 10–11

2.7. Local Mass Conservation.

  • The numerical flux balances the element source on every element, including boundary elements under the SIPG boundary jump and flux conventions.The local conservation law follows by testing with an elementwise constant function; the averaged normal flux is converted to the element's outward flux.Source: §2.7 · p. 11

3. Convergence Analysis

The convergence analysis combines SIPG stability and approximation with the reduced regularity of the measure-data adjoint and the control variational inequality. It develops interpolation and Ritz tools before deriving L2 error bounds for the control and state.

3. Convergence Analysis

  • The error analysis combines SIPG stability and approximation, reduced adjoint regularity from measure-valued tracking, and the variational inequality for pointwise control constraints.The analysis first collects DG auxiliary results and introduces interpolation and Ritz operators.Source: §3 · p. 11

3.1. Operators.

  • The analysis introduces the nodal interpolation operator Πh and the Ritz projection Rh associated with the SIPG bilinear form.Source: §3.1 · p. 12

3.1.1. The interpolation operator.

  • Global interpolation bounds control L2 and DG energy errors at order h^(1+α); point observations also require a maximum-norm estimate.Source: §3.1.1 · p. 13

3.1.2. The Ritz projection operator.

  • The Ritz-projection measure-norm estimate treats point, curve, and subdomain contributions separately; curve estimates use mesh-compatible observation sets.Source: §3.1.2 · pp. 13–14

3.2. Error Estimates.

  • Theorem 3.1 gives mesh-independent a priori L2 error bounds for the control and state, with θ = min{α, 2α−1} for α ∈ (1/2, 1].The control error is bounded first, and the state error is then recovered from it.Source: §3.2 · pp. 14, 16

4. Numerical Experiments

The experiments test convergence, local mass conservation, and tracking behavior across geometric observations and heterogeneous permeability configurations. The results support the predicted rates, show machine-precision conservation residuals, and indicate robust convergence across permeability jumps.

4.1. Numerical Implementation.

  • The implementation uses piecewise linear bases for state and adjoint, piecewise constants for control, and a primal-dual active set algorithm for the coupled system.The algorithm updates active and inactive control sets before solving the linear system.Source: §4.1 · p. 18

4.2. Example 1. Multi-feature geometric tracking.

  • Across three geometric-tracking cases, convergence orders stabilize at 0.97–1.22 from h = 2^-2 to h = 2^-5, and PDAS takes at most three iterations.The orders agree with the rate predicted by Theorem 3.1, while iteration counts do not grow with mesh refinement.Source: Table 1 · p. 19, §4.2 · p. 19
  • Across tracking cases, point weighting pulls pressure toward point targets, offset points yield asymmetric two-lobed controls, and line tracking yields symmetric flanking lobes.In the hybrid case, the weakly penalized subdomain pressure settles near its target without control saturation.Source: §4.2 · pp. 19–20, Figure 2 · p. 20

Numerical Verification of Local Mass Conservation.

4.3. Example 2. Heterogeneous domain with layers.

  • At finest refinement, heterogeneous cases achieve orders of 1.19–1.21 versus 1.21–1.23 for homogeneous cases, despite permeability jumps of 10 or 3 across the interface.The tracking region lies entirely on one side of the interface, so the contrast affects the flow path to the target.Source: §4.3 · pp. 21–22, Table 3 · p. 22

5. Conclusions

The paper develops a monolithic DG framework for Darcy optimal control with Radon-measure tracking and control constraints. Its analysis addresses reduced adjoint regularity, while experiments confirm predicted rates and reveal a sharper observed state-error rate than the proven bound.

5. Conclusions

  • The monolithic SIPG optimality system retains state, adjoint, and control as coupled unknowns, and inherits the DG scheme’s local mass conservation.The framework supports Radon-measure tracking with pointwise control constraints.Source: §5 · p. 22
  • An intermediate discrete adjoint driven by the continuous optimal state decouples adjoint and control errors, avoiding a circular argument in the analysis.The analysis also develops DG Ritz-projection estimates in the tracking-measure norm to address reduced adjoint regularity.Source: §5 · p. 22
  • Experiments confirm predicted rates and local conservation, while state errors appear to converge at O(h^2α), sharper than the proven bound; closing this gap remains future work.The paper also identifies time-dependent flow, more general permeability tensors, and adaptive meshes as future directions.Source: §5 · p. 22
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