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Divergence conforming DG method for the optimal control of the Oseen equation with variable viscosity

Harpal Singh, Arbaz Khan

TL;DR

The work develops a divergence-conforming discontinuous Galerkin method for PDE-constrained optimal control of stationary generalized Oseen flows with variable viscosity. It establishes rigorous a priori error estimates in energy norms and $L^2$-type norms, alongside new residual-based a posteriori error estimators with proven reliability and efficiency. A unified analysis covers diffusion- and convection-dominated regimes and yields robustness with respect to the Reynolds-like parameter $ u^{-1}$, applicable to Stokes, Brinkman, Oseen, and generalized Oseen models. Numerical experiments in both two and three dimensions confirm optimal convergence rates and the practical effectiveness of the estimators, including adaptive mesh refinement near boundary layers and geometric singularities.

Abstract

This study introduces the divergence-conforming discontinuous Galerkin finite element method (DGFEM) for numerically approximating optimal control problems with distributed constraints, specifically those governed by stationary generalized Oseen equations. We provide optimal a priori error estimates in energy norms for such problems using the divergence-conforming DGFEM approach. Moreover, we thoroughly analyze $L^2$ error estimates for scenarios dominated by diffusion and convection. Additionally, we establish the new reliable and efficient a posteriori error estimators for the optimal control of the Oseen equation with variable viscosity. Theoretical findings are validated through numerical experiments conducted in both two and three dimensions.

Divergence conforming DG method for the optimal control of the Oseen equation with variable viscosity

TL;DR

The work develops a divergence-conforming discontinuous Galerkin method for PDE-constrained optimal control of stationary generalized Oseen flows with variable viscosity. It establishes rigorous a priori error estimates in energy norms and -type norms, alongside new residual-based a posteriori error estimators with proven reliability and efficiency. A unified analysis covers diffusion- and convection-dominated regimes and yields robustness with respect to the Reynolds-like parameter , applicable to Stokes, Brinkman, Oseen, and generalized Oseen models. Numerical experiments in both two and three dimensions confirm optimal convergence rates and the practical effectiveness of the estimators, including adaptive mesh refinement near boundary layers and geometric singularities.

Abstract

This study introduces the divergence-conforming discontinuous Galerkin finite element method (DGFEM) for numerically approximating optimal control problems with distributed constraints, specifically those governed by stationary generalized Oseen equations. We provide optimal a priori error estimates in energy norms for such problems using the divergence-conforming DGFEM approach. Moreover, we thoroughly analyze error estimates for scenarios dominated by diffusion and convection. Additionally, we establish the new reliable and efficient a posteriori error estimators for the optimal control of the Oseen equation with variable viscosity. Theoretical findings are validated through numerical experiments conducted in both two and three dimensions.
Paper Structure (23 sections, 17 theorems, 147 equations, 16 figures)

This paper contains 23 sections, 17 theorems, 147 equations, 16 figures.

Table of Contents

  1. Introduction
  2. Optimal control problem
  3. Literature Review
  4. Main contribution:
  5. Outline of paper
  6. Preliminaries and continuous formulation
  7. Function spaces.
  8. Continuous formulation
  9. Second-order sufficient optimality conditions
  10. Discrete Formulation
  11. Discretization, Finite element spaces and traces
  12. $\boldsymbol{H}^{\text{div}}$-DG formulation for the Oseen problem with control
  13. A priori error estimates
  14. A posteriori error estimates
  15. Auxiliary forms and their properties.
  16. ...and 8 more sections

Key Result

Lemma 2.3

\newlabellem: 2.4.0 The Lagrangian $\mathcal{L}$, as defined in (2.16), is twice Fréchet-differentiable with respect to $\mathbf{v}=(\mathbf{y},\mathbf{u})$. The second-order derivative at $\mathbf{v} = (\tilde{\mathbf{y}},\tilde{\mathbf{u}})$ in conjunction with the associated adjoint state $\til for all $(\mathbf{s}_i,\mathbf{t}_i) \in \boldsymbol{V} \times \boldsymbol{L}^{2}(\Omega)$ where $i=

Figures (16)

  • Figure 1: Initial uniform meshes used for Examples-\ref{['Example 6.2.']}, and \ref{['Example 6.3.']}.
  • Figure 2: Plots of numerical solutions of state velocity $(\mathbf{y}_{h1},\mathbf{y}_{h2})$, co-state velocity $(\mathbf{w}_{h1},\mathbf{w}_{h2})$, state pressure $(p_h)$, co-state pressure $(r_h)$, and control $(\mathbf{u}_{h1},\mathbf{u}_{h2})$, respectively, for Example- \ref{['Example 6.1.']}.
  • Figure 3: Convergence plots for the state and co-state variables for Example- \ref{['Example 6.1.']}.
  • Figure 4: Convergence plot for the indicator and total error (uniform refinement) for Example- \ref{['Example 6.1.']}.
  • Figure 5: Adaptively refined meshes (a) 5784 DOF (b) 17380 DOF (c) 27948 DOF (showing the boundary layers).
  • ...and 11 more figures

Theorems & Definitions (38)

  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • Proof 1
  • Definition 2.4
  • Remark 2.5
  • Lemma 3.1
  • Proof 2
  • Lemma 4.1
  • Proof 3
  • ...and 28 more