A Pressure-Robust Embedded Discontinuous Galerkin Method for the Stokes Problem by Reconstruction Operators

Abstract

The embedded discontinuous Galerkin (EDG) finite element method for the Stokes problem results in a pointwise divergence-free approximate velocity on cells. However, the approximate velocity is not $H({div})$-conforming, and it can be shown that this is the reason that the EDG method is not pressure-robust, i.e., the error in the velocity depends on the continuous pressure. In this paper we present a local reconstruction operator that maps discretely divergence-free test functions to exactly divergence-free test functions. This local reconstruction operator restores pressure-robustness by only changing the right-hand side of the discretization, similar to the reconstruction operator recently introduced for the Taylor--Hood and mini elements by Lederer et al. [SIAM J. Numer. Anal., 55 (2017), pp. 1291--1314]. We present an a priori error analysis of the discretization showing optimal convergence rates and pressure-robustness of the velocity error. These results are verified by numerical examples. The motivation for this research is that the resulting EDG method combines the versatility of discontinuous Galerkin methods with the computational efficiency of continuous Galerkin methods and accuracy of pressure-robust finite element methods.