A hybridizable discontinuous Galerkin method for the incompressible Navier-Stokes equations

We present a hybridizable discontinuous Galerkin method for the numerical solution the incompressible Navier-Stokes equations. The method is devised by using the discontinuous Galerkin approximation with a special choice of the numerical traces and a fully implicit time-stepping method for temporal discretization. The HDG method possesses several unique features which distinguish themselves from other discontinuous Galerkin methods. First, it reduces the globally coupled unknowns to the approximate trace of the velocity and the mean of the pressure on element boundaries, thereby leading to a significant reduction in the degrees of freedom. Second, it allows for pressure, vorticity and stress boundary conditions to be prescribed on different parts of the boundary. Third, it provides, for smooth viscous-dominated problems, approximations of the velocity, pressure, and velocity gradient which converge with the optimal order of k+1 in the L²-norm, when polynomials of degree k = 0 are used for all components of the approximate solution. And fourth, it displays superconvergence properties that allow us to use the above-mentioned optimal convergence properties to define an element-by-element postprocessing scheme to compute a new and better approximate velocity. Indeed, this new approximation is exactly divergence-free, H(div)-conforming, and converges with order k + 2 for k ≥ 1 and with order 1 for k = 0 in the L ²-norm. We present extensive numerical results to demonstrate the accuracy and convergence properties of the method for a wide range of Reynolds numbers and for various polynomial degrees.