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A locally conservative LDG method for the incompressible Navier-Stokes equations. (English) Zbl 1069.76029
Summary: A new local discontinuous Galerkin (LDG) method for incompressible stationary Navier-Stokes equations is proposed and analyzed. Four important features render this method unique: its stability, its local conservativity, its high-order accuracy, and the exact satisfaction of the incompressibility constraint. Although the method uses completely discontinuous approximations, a globally divergence-free approximate velocity in \(H(\text{div};\Omega)\) is obtained by simple, element-by-element post-processing. Optimal error estimates are proven, and an iterative procedure used to compute the approximate solution is shown to converge. This procedure is nothing but a discrete version of classical fixed point iteration used to obtain existence and uniqueness of solutions to incompressible Navier-Stokes equations by solving a sequence of Oseen problems. Numerical results are shown which verify the theoretical rates of convergence. They also confirm the independence of the number of fixed point iterations with respect to the discretization parameters. Finally, they show that the method works well for a wide range of Reynolds numbers.

MSC:
76M10 Finite element methods applied to problems in fluid mechanics
76D05 Navier-Stokes equations for incompressible viscous fluids
65N15 Error bounds for boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
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