On setting a pressure datum when computing incompressible flows

SUMMARY The conventional approach to set the pressure level in a finite element discretization of an enclosed, steady, incompressible flow is to discard a continuity residual and set the associated pressure basis function coefficient to a desired value. Two issues surrounding this setting of a pressure datum are explored. First, it is shown that setting a boundary traction at a single node, in lieu of a Dirichlet velocity condition, is a preferred alternative for use with pressure-stabilized finite element methods. Second, it is shown that setting a pressure datum can slow or even stop the convergence of a GMRES-based iterative solver; though by some appearances a solution may appear to be converged, significant local errors in the velocity may exist. Under such circumstances it is preferable to solve the consistent singular system of equations, rather than setting a pressure datum. It is shown that GMRES converges in such cases, implicitly setting a pressure level that is determined from the initial guess. Copyright © 1999 John Wiley & Sons, Ltd. 1. BACKGROUND There has been a long history of research on proper and useful representations of the pressure field in finite element computations of incompressible flows. This prior research has resulted in conventional approaches involving mixed-order interpolation with the Galerkin finite element method (GFEM) and newer approaches employing equal-order interpolation with stabilized methods. Irrespective of the approach employed for computation, the solution of an incompressible flow within an enclosed domain contains a hydrostatic pressure mode that causes the pressure field to be indeterminant with respect to an arbitrary constant. Accommodating this simple effect is straightforward when employing classical GFEM techniques in conjunction with direct solution methods, but some important issues associated with the hydrostatic pressure mode when using newer finite element implementations, and when using iterative solution methods are discussed below. The relationship between the hydrostatic pressure mode and mass continuity of an incompressible fluid was clearly elucidated by Sani et al. [1,2] and Engelman et al. [3]. Following their work, our starting point is the statement of global mass conservation