This chapter discusses stabilized finite element formulations for incompressible flow computations. Finite element computation of incompressible flows involve two main sources of potential numerical instabilities associated with the Galerkin formulation of a problem. The stabilization techniques that are reviewed more extensively than others are the Galerkin/ least-squares (GLS), streamline-upwind/ Petrov–Galerkin (SUPG), and pressure-stabilizing/Petrov–Galerkin (PSPG) formulations. The SUPG stabilization for incompressible flows is achieved by adding to the Galerkin formulation a series of terms, each in the form of an integral over a different element. These integrals involve the product of the residual of the momentum equation and the advective operator acting on the test function. The natural boundary conditions are the conditions on the stress components, and these are the conditions assumed to be imposed at the remaining part of the boundary. The interpolation functions used for velocity and pressure are piecewise bilinear in space and piecewise linear in time. These computations involve no global coefficient matrices, and therefore need substantially less computer memory and time compared to noniterative solution of the fully discrete equations. It is suggested that for two-liquid flows, the solution and variational function spaces for pressure should include the functions that are discontinuous across the interface.
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