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A priori error estimates for stable generalized finite element discretization of parabolic interface optimal control problems

Xindan Zhang, Jianping Zhao, Yanren Hou

TL;DR

The paper tackles optimal control of a parabolic interface problem where the control acts on the interface and the diffusion coefficient jumps across the interface, causing reduced regularity. It develops a discretization framework using SGFEM for space, backward Euler for time, and variational discretization for the control, and proves a priori error estimates for the control, state, and adjoint state. The main result is a quantifiable convergence bound that captures the interplay between spatial mesh size $h$ and time step $\Delta t$, supported by numerical experiments across diverse interface geometries and coefficient contrasts. This conforming, parameter-free method is well-suited for moving and complex interfaces and demonstrates robust performance in practice.

Abstract

In this paper, we investigate optimal control problems governed by the parabolic interface equation, in which the control acts on the interface. The solution to this problem exhibits low global regularity due to the jump of the coefficient across the interface and the control acting on the interface. Consequently, the traditional finite element method fails to achieve optimal convergence rates when using a uniform mesh. To discretize the problem, we use fully discrete approximations based on the stable generalized finite element method for spatial discretization and the backward Euler scheme for temporal discretization, as well as variational discretization for the control variable. We prove a priori error estimates for the control, state, and adjoint state. Numerical examples are provided to support the theoretical findings.

A priori error estimates for stable generalized finite element discretization of parabolic interface optimal control problems

TL;DR

The paper tackles optimal control of a parabolic interface problem where the control acts on the interface and the diffusion coefficient jumps across the interface, causing reduced regularity. It develops a discretization framework using SGFEM for space, backward Euler for time, and variational discretization for the control, and proves a priori error estimates for the control, state, and adjoint state. The main result is a quantifiable convergence bound that captures the interplay between spatial mesh size and time step , supported by numerical experiments across diverse interface geometries and coefficient contrasts. This conforming, parameter-free method is well-suited for moving and complex interfaces and demonstrates robust performance in practice.

Abstract

In this paper, we investigate optimal control problems governed by the parabolic interface equation, in which the control acts on the interface. The solution to this problem exhibits low global regularity due to the jump of the coefficient across the interface and the control acting on the interface. Consequently, the traditional finite element method fails to achieve optimal convergence rates when using a uniform mesh. To discretize the problem, we use fully discrete approximations based on the stable generalized finite element method for spatial discretization and the backward Euler scheme for temporal discretization, as well as variational discretization for the control variable. We prove a priori error estimates for the control, state, and adjoint state. Numerical examples are provided to support the theoretical findings.
Paper Structure (8 sections, 12 theorems, 91 equations, 6 figures, 9 tables)

This paper contains 8 sections, 12 theorems, 91 equations, 6 figures, 9 tables.

Table of Contents

  1. Introduction
  2. Optimality system
  3. The Discretization of the Optimal Control Problem
  4. The stable generalized finite element method
  5. Fully discrete approximation of optimal control problems
  6. Error Analysis of Optimal Control Problems
  7. Numerical examples
  8. Conclusion

Key Result

Lemma 2.1

Assume that $f\in H^{1}(0,T;L^{2}(\Omega))$, $y_{0}\in H_{0}^{1}(\Omega)$ and $g+u\in L^{2}(0,T;H^{\frac{1}{2}}(\Gamma))$. Then there exists a unique solution

Figures (6)

  • Figure : Fig. 1. A geometry shape for the interface problem.
  • Figure : Fig. 2. An illustration of the enrichment nodes $P_i, i\in I_{enr}$.
  • Figure : Fig. 7. The error of the state, adjoint state, and control with $\beta^{-}/\beta^{+} = 1/1000$ for Example 1.
  • Figure : Fig. 12. The error of the state, adjoint state, and control (without control constraints) for Example 2.
  • Figure : Fig. 16. The error of the state, adjoint state, and control (with control constraints) for Example 2.
  • ...and 1 more figures

Theorems & Definitions (23)

  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • Lemma 2.5
  • proof
  • Remark 2.1
  • ...and 13 more