Mixed Discontinuous Galerkin Method for Quasi-Newtonian Stokes Flows (Journal of Computational Mathematics, 2024)
In this paper, we introduce and analyze an augmented mixed discontinuous Galerkin (MDG) method for a class of quasi-Newtonian Stokes flows. In the mixed formulation, the unknowns are strain rate, stress and velocity, which are approximated by a discontinuous piecewise polynomial triplet $\underline{\mathcal{P}}^S_{k+1}$-$\underline{\mathcal{P}}^S_{k+1}$-$\mathcal{P}_k$ for $k ≥ 0.$ Here, the discontinuous piecewise polynomial function spaces for the field of strain rate and the stress field are designed to be symmetric. In addition, the pressure is easily recovered through simple postprocessing. For the benefit of the analysis, we enrich the MDG scheme with the constitutive equation relating the stress and the strain rate, so that the well-posedness of the augmented formulation is obtained by a nonlinear functional analysis. For $k ≥ 0,$ we get the optimal convergence order for the stress in broken $\underline{H}$(div)-norm and velocity in $L^2$-norm. Furthermore, the error estimates of the strain rate and the stress in $\underline{L}^2$-norm, and the pressure in $L^2$-norm are optimal under certain conditions. Finally, several numerical examples are given to show the performance of the augmented MDG method and verify the theoretical results. Numerical evidence is provided to show that the orders of convergence are sharp.
Keywords: Quasi-Newtonian flows; Mixed discontinuous Galerkin method; Symmetric strain rate; Symmetric stress; Optimal convergence orders.