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An implicit high-order hybridizable discontinuous Galerkin method for the incompressible Navier-Stokes equations. (English) Zbl 1391.76353

Summary: We present an implicit high-order hybridizable discontinuous Galerkin method for the steady-state and time-dependent incompressible Navier-Stokes equations. The method is devised by using the discontinuous Galerkin discretization for a velocity gradient-pressure-velocity formulation of the incompressible Navier-Stokes equations with a special choice of the numerical traces. The method possesses several unique features which distinguish itself 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. Moreover, if the augmented Lagrangian method is used to solve the linearized system, the globally coupled unknowns become the approximate trace of the velocity only. Second, 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 \(L^{2}\)-norm, when polynomials of degree \(k\geqslant 0\) are used for all components of the approximate solution. And third, 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, \(H(\mathrm{div})\)-conforming, and converges with order \(k + 2\) for \(k\geqslant 1\) and with order 1 for \(k = 0\) in the \(L^{2}\)-norm. Moreover, a novel and systematic way is proposed for imposing boundary conditions for the stress, viscous stress, vorticity and pressure which are not naturally associated with the weak formulation of the method. This can be done on different parts of the boundary and does not result in the degradation of the optimal order of convergence properties of the method. Extensive numerical results are presented to demonstrate the convergence and accuracy properties of the method for a wide range of Reynolds numbers and for various polynomial degrees.

MSC:

76M10 Finite element methods applied to problems in fluid mechanics
76D05 Navier-Stokes equations for incompressible viscous fluids
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35Q30 Navier-Stokes equations
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