Pressure-robust $hp$-a posteriori error estimates of $\boldsymbol{H}(\mathrm{div})$-conforming discontinuous Galerkin methods for the Stokes equations
For researchers in numerical analysis of Stokes equations, this work provides rigorous a posteriori error estimators with pressure-robustness and hp-optimality, though the approach is incremental as it builds on existing techniques.
The paper develops pressure-robust residual-based hp-a posteriori error estimators for H(div)-conforming discontinuous Galerkin methods for the Stokes equations, providing upper and lower bounds for the velocity error that are independent of pressure and robust with respect to viscosity. Numerical results in 2D and 3D confirm reliability, efficiency, and pressure-robustness.
We devise and analyze a pressure-robust residual-based $hp$-a posteriori error estimator for $\boldsymbol{H}(\mathrm{div})$-conforming discontinuous Galerkin (dG) methods for the Stokes problem on two- and three-dimensional polytopal Lipschitz domains. The estimator provides an upper bound and a local lower bound for the velocity error in the energy norm, both robust with respect to the viscosity and independent of the pressure. Our analysis relies on a decomposition of the error into conforming and nonconforming parts. The nonconforming error is bounded using a partition-of-unity framework combined with local Helmholtz decompositions on vertex patches. The conforming error is analyzed by means of the generalized Bogovski\uı operator of [14] in both two and three dimensions, yielding two pressure-independent residual-based estimators associated with different interpolation operators. In the first approach, the upper bound for the conforming error consists of five error indicators and a data oscillation term. Four of these indicators exhibit $p$-optimal scaling, while the remaining one is suboptimal by a factor of $p^{1/2}$. In the second approach, the upper bound involves only two residual indicators together with the data oscillation term, at the expense of losing one order in $p$. Moreover, a pressure-robust local lower bound is established using $H^2$-bubble functions inspired by techniques developed for fourth-order PDEs. Numerical results in two and three dimensions confirm the reliability, efficiency, and pressure-robustness of the proposed estimators.