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A Posteriori Error Estimation for Pressure-Robust Finite Element Methods Applied to the Stokes Optimal Control Problem

Jingshi Li, Jiachuan Zhang

TL;DR

This work addresses a posteriori error estimation for pressure-robust finite element methods applied to the Stokes optimal control problem. By introducing a divergence-free reconstruction operator, the authors develop a three-term, pressure-independent a posteriori estimator for the control, state, and adjoint variables, built from a discretization of the optimality conditions and curl-based residuals that decouple velocity errors from pressure. They prove global reliability and efficiency, and validate the theory with 2D numerical experiments on uniform and adaptive meshes, demonstrating mesh-independent accuracy measures and near-optimal convergence. The results enhance the robustness and reliability of simulations for incompressible flow control, particularly in regimes with small viscosity or large pressure, by ensuring velocity error estimates are not polluted by the pressure term.

Abstract

This paper study a posteriori error estimates for the pressure-robust finite element method, which incorporates a divergence-free reconstruction operator, within the context of the distributed optimal control problem constrained by the Stokes equations. We develop an enhanced residual-based a posteriori error estimator that is independent of pressure and establish its global reliability and efficiency. The proposed a posteriori error estimator enables the separation of velocity and pressure errors in a posteriori error estimation, ensuring velocity-related estimates are free of pressure influence. Numerical experiments confirm our conclusions.

A Posteriori Error Estimation for Pressure-Robust Finite Element Methods Applied to the Stokes Optimal Control Problem

TL;DR

This work addresses a posteriori error estimation for pressure-robust finite element methods applied to the Stokes optimal control problem. By introducing a divergence-free reconstruction operator, the authors develop a three-term, pressure-independent a posteriori estimator for the control, state, and adjoint variables, built from a discretization of the optimality conditions and curl-based residuals that decouple velocity errors from pressure. They prove global reliability and efficiency, and validate the theory with 2D numerical experiments on uniform and adaptive meshes, demonstrating mesh-independent accuracy measures and near-optimal convergence. The results enhance the robustness and reliability of simulations for incompressible flow control, particularly in regimes with small viscosity or large pressure, by ensuring velocity error estimates are not polluted by the pressure term.

Abstract

This paper study a posteriori error estimates for the pressure-robust finite element method, which incorporates a divergence-free reconstruction operator, within the context of the distributed optimal control problem constrained by the Stokes equations. We develop an enhanced residual-based a posteriori error estimator that is independent of pressure and establish its global reliability and efficiency. The proposed a posteriori error estimator enables the separation of velocity and pressure errors in a posteriori error estimation, ensuring velocity-related estimates are free of pressure influence. Numerical experiments confirm our conclusions.

Paper Structure

This paper contains 9 sections, 7 theorems, 69 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Model problem and notations
  3. The optimal control problem constrained by the Stokes equations
  4. Notation
  5. Classical finite element discretization and standard a posteriori error estimates
  6. Pressure-robust finite element discretization and improved a posteriori error estimates
  7. Reconstruction operator and existing pressure-robust results for the Stokes equations
  8. Pressure-robust approximation and improved a posteriori error estimation for the optimal control problem
  9. Numerical example

Key Result

Lemma 2.1

(see Theorem 5.4 in girault2012finite) If $\textit{f}\in \boldsymbol{U}$, and $(\textbf{w},\rho)$ is the solution of the following equation Then, it has where $C$ is independent of $\textbf{w}, \rho, \textit{f}$.

Figures (4)

  • Figure 5.1: Reference solutions for $\boldsymbol{y}$ (left), $\boldsymbol{z}$ (middle), and $\boldsymbol{u}$ (right) of (\ref{['Eq: opt sys']}) with $d.o.f=2363396$ for $(\nu,\alpha)=(1,10^{-6})$ in top row and $(\nu,\alpha)=(10^{-2},10^{-4})$ in bottom row in Example 1.
  • Figure 5.2: Convergence rates (top row) and its individual contributions (bottom row) with $(\nu,\alpha)=(1,10^{-6})$ on the left and $(\nu,\alpha)=(10^{-2},10^{-4})$ on the right in Example 1.
  • Figure 5.3: The convergent rates for Example 2.
  • Figure 5.4: The initial mesh (left) and the seventh refinement mesh (right) for Example 2

Theorems & Definitions (14)

  • Lemma 2.1
  • Lemma 2.2
  • Remark 2.1
  • Theorem 3.1
  • Remark 3.1
  • Theorem 4.1: see ledererRefinedPosterioriError2019
  • Remark 4.1
  • Remark 4.2
  • Lemma 4.1
  • proof
  • ...and 4 more