A Posteriori Error Estimation Improved by a Reconstruction Operator for the Stokes Optimal Control Problem

This paper focuses on a posteriori error estimates for a 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.

💡 Research Summary

This paper addresses a critical gap in the a‑posteriori error analysis of distributed optimal control problems governed by the Stokes equations. While a‑priori error estimates for pressure‑robust finite‑element discretizations have been developed, existing a‑posteriori estimators still couple velocity and pressure errors. Such coupling becomes especially problematic for low viscosity (small ν) or large pressure fields, where the velocity error is amplified by a factor of ν⁻¹ ‖p‖, leading to loss of robustness and possible numerical instability.

The authors adopt the third of three known strategies for achieving pressure robustness: the use of a divergence‑free reconstruction operator (R_h). This operator maps the discrete velocity space (V_h) into an (H(\mathrm{div}))‑conforming Raviart–Thomas space (W_h) while satisfying three key properties: (i) the reconstructed field is exactly divergence‑free and has zero normal trace on the boundary; (ii) it is (L^2)‑orthogonal to piecewise constants on each element; (iii) the reconstruction error scales with the mesh size (h) and the local Sobolev seminorms of the original function. The existence of such an operator is guaranteed by prior work (Linke, Lederer, et al.).

With (R_h) in hand, the right‑hand side of the Stokes momentum equation is modified from ((f,v_h)) to ((f,R_h v_h)). This modification preserves the continuous problem’s invariance under the addition of a gradient to the pressure, thereby rendering the discrete scheme pressure‑robust: the discrete velocity solution no longer depends on the pressure gradient term.

Building on this pressure‑robust discretization, the paper derives a new residual‑based a‑posteriori error estimator (\widehat{\varepsilon}_{\mathrm{PR}}). The estimator consists of element‑wise volume contributions that involve only the reconstructed residuals \