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An Error Analysis of Second Order Elliptic Optimal Control Problem via Hybrid Higher Order Methods

Gouranga Mallik, Ramesh Chandra Sau

TL;DR

The paper develops Hybrid High-Order (HHO) methods for distributed optimal control of the Poisson equation, proposing three unconstrained schemes (UC1, UC2, UC3.1/UC3.2) and two constrained schemes (WC1, WC2). By leveraging a reconstruction-based approach, UC3.1/UC3.2 achieve improved control accuracy (up to \\(k+2\\) order) compared to standard FEM, while WC1 attains linear convergence with minimal element order and WC2 attains cubic convergence for the control under suitable regularity. The analysis combines the HHO framework with variational discretization concepts, providing rigorous energy and L2-error estimates for states, adjoints, and controls. Numerical experiments on rectangular and polygonal meshes validate the theoretical rates and demonstrate the practical robustness and efficiency of the proposed schemes. The results position HHO methods as powerful tools for high-accuracy optimal control in polytopal meshes, with clear advantages over traditional discretizations.

Abstract

This paper presents the design and analysis of a Hybrid High-Order (HHO) approximation for a distributed optimal control problem governed by the Poisson equation. We propose three distinct schemes to address unconstrained control problems and two schemes for constrained control problems. For the unconstrained control problem, while standard finite elements achieve a convergence rate of \( k+1 \) (with \( k \) representing the polynomial degree), our approach enhances this rate to \( k+2 \) by selecting the control from a carefully constructed reconstruction space. For the box-constrained problem, we demonstrate that using lowest-order elements (\( \mathbb{P}_0 \)) yields linear convergence, in contrast to finite element methods (FEM) that require linear elements to achieve comparable results. Furthermore, we derive a cubic convergence rate for control in the variational discretization scheme. Numerical experiments are provided to validate the theoretical findings.

An Error Analysis of Second Order Elliptic Optimal Control Problem via Hybrid Higher Order Methods

TL;DR

The paper develops Hybrid High-Order (HHO) methods for distributed optimal control of the Poisson equation, proposing three unconstrained schemes (UC1, UC2, UC3.1/UC3.2) and two constrained schemes (WC1, WC2). By leveraging a reconstruction-based approach, UC3.1/UC3.2 achieve improved control accuracy (up to \ order) compared to standard FEM, while WC1 attains linear convergence with minimal element order and WC2 attains cubic convergence for the control under suitable regularity. The analysis combines the HHO framework with variational discretization concepts, providing rigorous energy and L2-error estimates for states, adjoints, and controls. Numerical experiments on rectangular and polygonal meshes validate the theoretical rates and demonstrate the practical robustness and efficiency of the proposed schemes. The results position HHO methods as powerful tools for high-accuracy optimal control in polytopal meshes, with clear advantages over traditional discretizations.

Abstract

This paper presents the design and analysis of a Hybrid High-Order (HHO) approximation for a distributed optimal control problem governed by the Poisson equation. We propose three distinct schemes to address unconstrained control problems and two schemes for constrained control problems. For the unconstrained control problem, while standard finite elements achieve a convergence rate of (with representing the polynomial degree), our approach enhances this rate to by selecting the control from a carefully constructed reconstruction space. For the box-constrained problem, we demonstrate that using lowest-order elements () yields linear convergence, in contrast to finite element methods (FEM) that require linear elements to achieve comparable results. Furthermore, we derive a cubic convergence rate for control in the variational discretization scheme. Numerical experiments are provided to validate the theoretical findings.
Paper Structure (26 sections, 29 theorems, 165 equations, 9 figures, 3 tables)

This paper contains 26 sections, 29 theorems, 165 equations, 9 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Preliminaries on HHO method
  3. Discrete setting for HHO method.
  4. Discrete spaces
  5. Norms
  6. Local and global reduction operators
  7. Local reconstruction and stabilization operators.
  8. Local elliptic projection operator
  9. Bilinear forms
  10. Trace inequality and estimate for projection
  11. Integration by parts formula
  12. Optimal control problem without control constraints
  13. Existence, uniqueness, and regularity for \ref{['eqn:conti_costfun']}-\ref{['eqn:conti_state']}
  14. Scheme UC1
  15. Scheme UC2 (Variational Discretization):
  16. ...and 11 more sections

Key Result

Lemma 3.1

For $f,y_d\in H^k(\mathcal{T}_h)$ with $k\geq 0,$ the solution of eqn:conti_os_state-eqn:conti_os_control has the regularity $y,u, \phi \in H^{k+2}(\mathcal{T}_h).$

Figures (9)

  • Figure 5.1: Meshes
  • Figure 5.2: Convergence results for Scheme UC1 & UC2 on rectangular mesh.
  • Figure 5.3: Convergence results for Scheme UC1 & UC2 on polygonal mesh.
  • Figure 5.4: Convergence results for Scheme UC3.1 on rectangular mesh.
  • Figure 5.5: Convergence results for Scheme UC3.1 on polygonal mesh.
  • ...and 4 more figures

Theorems & Definitions (48)

  • Remark 2.1
  • Lemma 3.1: Regularity
  • proof
  • Theorem 3.2: Existence and uniqueness of discrete solution
  • proof
  • Theorem 3.3: Abstract $L^2$- error estimates of control
  • proof
  • Lemma 3.4
  • proof
  • Lemma 3.5
  • ...and 38 more