Local projection stabilized Lagrange-Galerkin methods for Navier-Stokes equations at high Reynolds numbers (Q2321000)

The paper presents Lagrange-Galerkin methods for the incompressible Navier-Stokes equations. A discrete Galerkin method with finite elements is combined with a backward in time discretization along the characteristics of the convection terms. Upwinding is thus introduced by discretizing the material derivative. Moreover, a local projection stabilization is applied with inf-sup conforming finite elements. Therewith convection-dominated diffusion problems can be handled and also high Reynolds number flows. The method is shown to be unconditionally stable given that all integrals could be computed exactly. The convergence order of the scheme in space and time, i.e., \(m\)-th order (corresponding to the order of the elements) and second order, respectively, is proven. Numerical examples illustrate the stability of the method and show that it is more stable than the conventional Lagrange-Galerkin methods. Numerical results are given for a turbulent flow past a NACA 0012 airfoil at zero angle attack, and for the 3D lid-driven cavity benchmark problem.