Pressure Stability in Fractional Step Finite Element Methods for Incompressible Flows

The objective of this paper is to analyze the pressure stability of fractional step finite element methods for incompressible flows that use a pressure Poisson equation. For the classical first-order projection method, it is shown that there is a pressure control which depends on the time step size, and therefore there is a lower bound for this time step for stability reasons. The situation is much worse for a second-order scheme in which part of the pressure gradient is kept in the momentum equation. The pressure stability in this case is extremely weak. To overcome these shortcomings, a stabilized fractional step finite element method is also considered, and its stability is analyzed. Some simple numerical examples are presented to support the theoretical results.