Pressure-robust $hp$-a posteriori error estimates of $\boldsymbol{H}(\mathrm{div})$-conforming discontinuous Galerkin methods for the Stokes equations

Abstract

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.