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

TL;DR

This paper develops and analyzes a stabilized finite element scheme for an optimal control problem governed by a parabolic convection--diffusion equation, providing error estimates and numerical validation.

Contribution

It introduces a stabilized algebraic flux correction method for discretizing the control problem and proves existence, uniqueness, and convergence of the scheme.

Findings

Error estimates in various norms for state, co-state, and control variables.

The scheme is stable and convergent under mild mesh and time step conditions.

Numerical experiments confirm theoretical convergence rates and effectiveness in convection-dominant scenarios.

Abstract

We consider an optimal control problem on a bounded domain $\Omega\subset\mathbb{R}^2,$ 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.