HDG Methods for Stokes Equation Based on Strong Symmetric Stress Formulations

Abstract

We propose a hybridizable discontinuous Galerkin (HDG) method for Stokes equation based on strong symmetric stress formulations. Stress-based method stands out compared to gradient or vorticity-based methods by its simple and natural way of enforcing Neumann type boundary conditions. Our method uses the polynomial spaces of orders $${\text {k}}-({\text {k}}+1)-{\text {k}}$$ for the stress–velocity–pressure triplet, and orders $${\text {k}}-({\text {k}}+1)$$ for the tangential and the normal components of the numerical trace space. By sending the normal stabilization to infinity, we obtain another method that provides exactly divergence-free solution and is pressure-robust. We prove that both methods are optimal for all variables and achieve super-convergence for the numerical trace. In addition, we build a quantitative connection between the normal stabilization and the pressure-robustness. Numerical experiments are also presented to validate our theoretical discoveries.