p=2 Continuous finite elements on tetrahedra with local mass matrix inversion to solve the preconditioned compressible Navier-Stokes equations

For unsteady simulations a global mass matrix must be inverted when using a continuous finite element method. To avoid this, a local approximate mass matrix inversion procedure is given for a p=2 basis. This inversion process when combined with a low Mach Turkel preconditioner, dual time stepping, and multigrid gives an efficient and accurate iterative scheme to solve the compressible Navier-Stokes equations. The performance of the preconditioner for both p=1 and p=2 is examined on a perturbed free stream flow under a wide range of flow conditions. Both preconditioning and multigrid improved the convergence rates for all cases and for both polynomial degrees the performance is similar in most cases. The best performance is seen for unsteady and viscous flow problems. In these regimes the iterative convergence is almost independent of the polynomial degree and it takes around 1-3 multigrid cycles to reduce the residual one order of magnitude. A large scale simulation of flow around a NACA 0012 airfoil is also presented. For the flow conditions of M=0.1,Re=500, and CFL=55 the convergence rate averaged 2.5 cycles per decade. Also, the method showed good linear speedup when solved in parallel, as a speedup of 160 was achieved with 192 processors.