Fourier Analysis of an Equal Order Incompressible Flow Solver Stabilized by Pressuregradient

Abstract

Fourier analysis techniques are applied to the stabilized finite element method recently proposed by Codina and Blasco
for the approximation of the incompressible Navier-Stokes equations, here denoted by SPGP method (Stabilization by Pressure Gradient Projection). The analysis is motivated by spurious waves that pollute the computed pressure in start-up
ows simulation. An example of this spurious phenomenon is reported. It is shown that Fourier techniques can predict the numerical behavior of stabilized methods with remarkable accuracy, even though the original Navier-Stokes setting must be significantly simplified to apply them. In the steady case good estimates for the stabilization parameters are obtained. In the transient case spurious long waves are shown to be persistent when the element Reynolds number is large and the Courant number is small. This can be avoided by treating the pressure gradient projection implicitly, though with additional computing effort. Standard extrapolation variants are unfortunately unstable. Comparisons to Galerkin-Least-Squares method and Chorin's projection method are also addressed.