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Error estimates of $hp$-finite element method for elliptic optimal control problems with robin boundary

Xingyuan Lin, Xiuxiu Lin, Xuesong Chen

TL;DR

The paper develops a rigorous error analysis for the $hp$-finite element method applied to elliptic optimal control with Robin boundary conditions and boundary observations. It derives a priori error estimates using a Clément-type interpolation and an auxiliary system, yielding rates that depend on the local mesh size and polynomial degree, specifically with $ ig orm{u-u_{hp}}_{L^2( ext{Ω})} + ig orm{y-y_{hp}}_{H^1( ext{Ω})} + ig orm{z-z_{hp}}_{H^1( ext{Ω})} \\le C rac{h^{ u-1}}{p^{m-1}} ig( orm{y}_{H^{m}( ext{Ω})} + orm{z}_{H^{m}( ext{Ω})}ig)$ where $ u= ext{min}\{p+1,motig)$. It then constructs a computable, residual-based a posteriori error estimator using Scott–Zhang interpolation, proves reliability and efficiency, and validates the theory with numerical experiments showing effective adaptive refinement and agreement between estimated and true errors. The results broaden the applicability of $hp$-FEM to Robin-type boundary control problems and offer practical guidance for adaptive strategies in PDE-constrained optimization.

Abstract

A priori and a posteriori error analysis of $hp$ finite element method for elliptic control problem with Robin boundary condition and boundary observation are presented. are presented. Through the Clément-type approach and the construction of an auxiliary system, we derived a priori error estimates for the elliptic optimal control problem. Residual-based a posteriori error estimates are derived based on the well-known Scott-Zhang-type quasi-interpolation and coupled state-control approximations, thus establishing an a posteriori error estimator for the $hp$ finite element method. The numerical example demonstrates the accuracy of error estimation for the elliptic optimal control problems with Robin boundary.

Error estimates of $hp$-finite element method for elliptic optimal control problems with robin boundary

TL;DR

The paper develops a rigorous error analysis for the -finite element method applied to elliptic optimal control with Robin boundary conditions and boundary observations. It derives a priori error estimates using a Clément-type interpolation and an auxiliary system, yielding rates that depend on the local mesh size and polynomial degree, specifically with where . It then constructs a computable, residual-based a posteriori error estimator using Scott–Zhang interpolation, proves reliability and efficiency, and validates the theory with numerical experiments showing effective adaptive refinement and agreement between estimated and true errors. The results broaden the applicability of -FEM to Robin-type boundary control problems and offer practical guidance for adaptive strategies in PDE-constrained optimization.

Abstract

A priori and a posteriori error analysis of finite element method for elliptic control problem with Robin boundary condition and boundary observation are presented. are presented. Through the Clément-type approach and the construction of an auxiliary system, we derived a priori error estimates for the elliptic optimal control problem. Residual-based a posteriori error estimates are derived based on the well-known Scott-Zhang-type quasi-interpolation and coupled state-control approximations, thus establishing an a posteriori error estimator for the finite element method. The numerical example demonstrates the accuracy of error estimation for the elliptic optimal control problems with Robin boundary.
Paper Structure (8 sections, 10 theorems, 86 equations, 4 figures, 6 tables, 1 algorithm)

This paper contains 8 sections, 10 theorems, 86 equations, 4 figures, 6 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. $hp$ finite element approximation and preliminary results
  3. A priori error estimate
  4. A posterior error estimate and algorithm
  5. A posterior error estimate
  6. Numerical algorithm
  7. Numerical experiments
  8. Conclusion

Key Result

Lemma 2.1

(see troeltzsch2010optimal) The control problem change_problem has a unique solution $(y,u)\in Y\times U$ if and only if there exists $z\in Y$ such that the triplet $(u,y,z)$ satisfies the following optimality conditions:

Figures (4)

  • Figure 1: Numerical solution for $N=64$ and $\mathbf{p}=2$
  • Figure 2: Error for $N=64$ and $\mathbf{p}=2$
  • Figure 3: Numerical solution for $N=64$ and $\mathbf{p}=2$
  • Figure 4: Error for $N=64$ and $\mathbf{p}=2$

Theorems & Definitions (11)

  • Lemma 2.1
  • Lemma 2.2
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Theorem 3.1
  • Theorem 3.2
  • Lemma 4.1
  • Lemma 4.2
  • Definition 4.1
  • ...and 1 more