Adaptive $H$(div)-Conforming Embedded-Hybridized Discontinuous Galerkin Finite Element Methods for the Stokes Problems (CSIAM Transactions on Applied Mathematics, 2022)
In this paper, we propose a residual-based a posteriori error estimator ofembedded–hybridized discontinuous Galerkin finite element methods for the Stokesproblems in two and three dimensions. The piecewise polynomials of degree $k (k≥1)$ and $k−1$ are used to approximate the velocity and pressure in the interior of elements, and the piecewise polynomials of degree $k$ are utilized to approximate the velocity and pressure on the inter-element boundaries. The attractive properties, nameddivergence-free and $H$(div)-conforming, are satisfied by the approximate velocity field.We prove that the a posteriori error estimator is robust in the sense that the ratio of theupper and lower bounds is independent of the mesh size and the viscosity. Finally, weprovide several numerical examples to verify the theoretical results.
Keywords: Stokes equations; HDG methods; E-HDG methods; a posteriori error estimator; divergence-free; $H$(div)-conforming.