$$H(\textrm{div})$$ -conforming HDG methods for the stress-velocity formulation of the Stokes equations and the Navier–Stokes equations

Abstract

In this paper we devise and analyze a pressure-robust and superconvergent HDG method in stress-velocity formulation for the Stokes equations and the Navier–Stokes equations with strongly symmetric stress. The stress and velocity are approximated using piecewise polynomial space of order k and \(H(\textrm{div};\Omega )\)-conforming space of order \(k+1\), respectively, where k is the polynomial order. In contrast, the tangential trace of the velocity is approximated using piecewise polynomials of order k. Moreover, the characterization of the proposed schemes shows that the globally coupled unknowns are the normal trace and the tangential trace of velocity, and the piecewise constant approximation for the trace of the stress. The discrete \(H^1\)-stability is established for the discrete solution. The proposed formulation yields divergence-free velocity, but causes difficulties for the derivation of the pressure-independent error estimate given that the pressure variable is not employed explicitly in the discrete formulation. This difficulty can be overcome by observing that the \(L^2\) projection to the stress space has a nice commuting property. Moreover, superconvergence for velocity in discrete \(H^1\)-norm is obtained, with regard to the degrees of freedom of the globally coupled unknowns. Then the convergence of the discrete solution to the weak solution for the Navier–Stokes equations via the compactness argument is rigorously analyzed under minimal regularity assumption. The strong convergence for velocity and stress is proved. Importantly, the strong convergence for velocity in discrete \(H^1\)-norm is achieved. Several numerical experiments are carried out to confirm the proposed theories.