Numerical analysis of a stabilized scheme for an optimal control problem governed by a parabolic convection--diffusion equation

TL;DR

The paper tackles an optimal control problem governed by a parabolic convection–diffusion–reaction equation with pointwise control bounds. It employs an optimize-then-discretize strategy, finite element spatial discretization with algebraic flux correction (AFC) stabilization, and backward Euler time stepping, with control discretization realized via variational discretization. A nonlinear AFC fully discrete scheme is analyzed for existence and uniqueness under a mild time-step to mesh-size relation (k = O(h)), and rigorous error estimates are derived for the state, co-state, and control in appropriate norms. Numerical experiments validate the theoretical convergence rates and demonstrate robustness in convection-dominated regimes and interior-layer settings, highlighting the practical viability of AFC stabilization in parabolic optimal control problems.

Abstract

We consider an optimal control problem on a bounded domain $Ω\subset\mathbb{R}^2,$ governed by a parabolic convection--diffusion--reaction equation with pointwise control constraints. We follow the optimize--then--discretize approach, in which the state and co-state variables are discretized using the piecewise linear finite element method. For stabilization, we apply the algebraic flux correction method. Temporal discretization is performed using the backward Euler method. The discrete control variable is obtained by projecting the discretized adjoint state onto the set of admissible controls. The resulting stabilized fully--discrete scheme is nonlinear and a fixed point argument is used to prove its existence and uniqueness under a mild condition between the time step $k$ and the mesh size $h,$ e.g., $k = \mathcal{O}(h).$ Furthermore, assuming sufficient regularity of the exact solution, we derive error estimates in the $L^{2}$ and energy norms with respect to the spatial variable, and in the $\ell^\infty$ norm with respect to time for the state and co-state variables. For the control variable, we also derive an $L^{2}$-norm error estimate with respect to space and an $\ell^\infty$-norm estimate in time. Finally, we present numerical experiments that validate the the order of convergence of the stabilized fully--discrete scheme based on the algebraic flux correction method. We also test the stabilized fully--discrete scheme in optimal control problems that governed by a convection--dominant equation where the solution possesses interior layers.

Figures (5)

• Figure 2.1: The subdomain $\omega_i$ of the triangulation $\mathcal{T}_h.$
• Figure 5.1: A triangulation of a square domain.
• Figure 5.2: Results using AFC: (top) state and co-state for $\mu = 10^{-8}$, (bottom) convergence studies for $\mu = 10^{-4},\, 10^{-5},\,10^{-8}$.
• Figure 5.3: Results using AFC: (top) state and co-state for $\mu = 10^{-4}$, (bottom) convergence studies for $\mu = 10^{-4}$ and $\mu = 10^{-5}$.
• Figure 5.4: Results using AFC: (top) state and co-state for $\mu = 10^{-4}$, (bottom) convergence studies for $\mu = 10^{-4}$ and $\mu = 10^{-5}$.

Theorems & Definitions (52)

• Remark 2.1
• Lemma 2.1
• Remark 2.2
• Remark 2.3
• Remark 2.4
• Remark 2.5
• Remark 2.6
• Remark 2.7
• Definition 2.1
• Remark 2.8