Quadratic finite elements and incompressible viscous flows

Pressure stabilization methods are applied to higher-order velocity finite elements for application to viscous incompressible flows. Both a standard pressure stabilizing Petrov–Galerkin (PSPG) method and a new polynomial pressure projection stabilization (PPPS) method have been implemented and tested for various quadratic elements in two dimensions. A preconditioner based on relaxing the incompressibility constraint is also tested for the iterative solution of saddle point problems arising from mixed Galerkin finite element approximations to the Navier–Stokes equations. The preconditioner is demonstrated for BB stable elements with discontinuous pressure approximations in two and three dimensions.