This paper develops a novel residual-based a posteriori error estimator for a pressure-robust Galerkin finite-element method for the Stokes equations. Recent studies show that pressure-robust Stokes solvers outperform non-robust solvers, especially for problems with small viscosity or singular pressure unknowns. We base our analysis on a pressure-robust scheme introduced in [22], which approximates velocity with piecewise continuous \(P_1\) functions enriched by RT \(_0\) functions and pressure with discontinuous \(P_0\) functions. While previous a priori error estimates show velocity error independence from pressure and viscosity, we propose an efficient and reliable a posteriori error estimator that maintains this robustness. Theoretical analysis demonstrates the equivalence between the exact \(H^1\) velocity error and the fully computable proposed estimator. Unlike traditional a posteriori estimates that include pressure contributions, our approach depends solely on velocity-related terms. This enables the estimator to distinguish between velocity and pressure singularities and guide mesh refinement for velocity improvement. Numerical results validate the efficiency and reliability of the estimate, and the developed error indicator is applied in an automatic, self-adaptive mesh refinement process to capture velocity singularities.
