Stabilization of incompressibility and convection through orthogonal sub-scales in finite element methods

Two apparently different forms of dealing with the numerical instability) due to the incompressibility constraint of the Stokes problem are analyzed in this paper. The first of them is the stabilization through the pressure gradient projection. which consists of adding a certain least-squares form of the difference between the pressure gradient and its L-2 projection onto the discrete velocity space in the variational equations of the problem. The second is a sub-grid scale method, whose stabilization effect is very similar to that of the Galerkin/least-squares (GLS) method for the Stokes problem. It is shown here that the first method can also be recast in the framework of sub-grid scale methods with a particular choice for the space of sub-scales. This leads to a new stabilization procedure, whose applicability to stabilize convection is also studied in this paper. (C) 2000 Elsevier Science B.V. All rights reserved.