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A monotone finite element method for an elliptic distributed optimal control problem with a convection-dominated state equation

SeongHee Jeong, Seulip Lee, Sijing Liu

TL;DR

The paper addresses an elliptic distributed optimal control problem constrained by a convection-diffusion-reaction state equation in the convection-dominated regime. It introduces a monotone discretization based on the edge-averaged finite element (EAFE) scheme, which preserves the discrete maximum principle and the desired-state bounds, ensuring oscillation-free and stable solutions. A rigorous analysis combines EAFE consistency with a discrete inf-sup condition to establish well-posedness and first-order convergence, supported by comprehensive numerical experiments that validate stability and accuracy in boundary- and interior-layer scenarios. The work provides a robust numerical framework for convection-dominated OCPs and lays groundwork for adaptive refinement and multiphysics extensions.

Abstract

We propose and analyze a monotone finite element method for an elliptic distributed optimal control problem constrained by a convection-diffusion-reaction equation in the convection-dominated regime. The method is based on the edge-averaged finite element (EAFE) scheme, which is known to preserve the discrete maximum principle for convection-diffusion problems. We show that the EAFE discretization inherits the monotonicity property of the continuous problem and consequently preserves the desired-state bounds at the discrete level, ensuring that the numerical optimal state remains stable and free of nonphysical oscillations. The discrete formulation is analyzed using a combination of the EAFE consistency result and a discrete inf-sup condition, which together guarantee well-posedness and yield the optimal convergence order. Comprehensive numerical experiments are presented to confirm the theoretical findings and to demonstrate the robustness of the proposed scheme in the convection-dominated regimes.

A monotone finite element method for an elliptic distributed optimal control problem with a convection-dominated state equation

TL;DR

The paper addresses an elliptic distributed optimal control problem constrained by a convection-diffusion-reaction state equation in the convection-dominated regime. It introduces a monotone discretization based on the edge-averaged finite element (EAFE) scheme, which preserves the discrete maximum principle and the desired-state bounds, ensuring oscillation-free and stable solutions. A rigorous analysis combines EAFE consistency with a discrete inf-sup condition to establish well-posedness and first-order convergence, supported by comprehensive numerical experiments that validate stability and accuracy in boundary- and interior-layer scenarios. The work provides a robust numerical framework for convection-dominated OCPs and lays groundwork for adaptive refinement and multiphysics extensions.

Abstract

We propose and analyze a monotone finite element method for an elliptic distributed optimal control problem constrained by a convection-diffusion-reaction equation in the convection-dominated regime. The method is based on the edge-averaged finite element (EAFE) scheme, which is known to preserve the discrete maximum principle for convection-diffusion problems. We show that the EAFE discretization inherits the monotonicity property of the continuous problem and consequently preserves the desired-state bounds at the discrete level, ensuring that the numerical optimal state remains stable and free of nonphysical oscillations. The discrete formulation is analyzed using a combination of the EAFE consistency result and a discrete inf-sup condition, which together guarantee well-posedness and yield the optimal convergence order. Comprehensive numerical experiments are presented to confirm the theoretical findings and to demonstrate the robustness of the proposed scheme in the convection-dominated regimes.

Paper Structure

This paper contains 18 sections, 4 theorems, 78 equations, 3 figures, 8 tables.

Table of Contents

  1. Introduction
  2. Continuous Problems and Preliminaries
  3. Optimal Control Problems
  4. Maximum Principle and Stability via Desired-State Bounds
  5. A Monotone Scheme for Optimal Control Problems
  6. Edge-Averaged Finite Element Discretization
  7. Exponentially Fitted Flux and Its Approximation
  8. Edge-Averaged Finite Element (EAFE) Scheme
  9. A Monotone Scheme for Optimal Control Problems
  10. Preservation of Desired-State Bounds
  11. Error Analysis
  12. Well-Posedness of the Discrete Problem
  13. Convergence Results
  14. Numerical Experiments
  15. Stability Test (Desired-State Bounds)
  16. ...and 3 more sections

Key Result

Lemma 3.1

Assume that $\varepsilon\in W^{1,\infty}(T)$ and $\bm{\zeta}\in [W^{1,\infty}(T)]^2$ for all $T\in\mathcal{T}_h$. Then, there exists a constant $C>0$, independent of $h$, such that A proof of this estimate can be found in xu1999monotone.

Figures (3)

  • Figure 1: Optimal and adjoint states for Example \ref{['ex:dsbound']} with $\varepsilon = 10^{-9}$.
  • Figure 2: Numerical solution and exact solution for Example \ref{['ex:bdlayer']} with $\varepsilon=10^{-9}$
  • Figure 3: Numerical solution and exact solution for Example \ref{['ex:inlayer']} with $\varepsilon=10^{-9}$

Theorems & Definitions (17)

  • Remark 2.1
  • Lemma 3.1: Consistency estimate
  • Remark 3.2
  • Remark 3.3
  • Remark 3.4
  • Remark 3.5
  • Theorem 3.6
  • proof
  • Remark 3.7
  • Lemma 4.1
  • ...and 7 more