The cause and cure (!) of the spurious pressures generated by certain fem solutions of the incompressible Navier‐Stokes equations: Part 2

Abstract

SUMMARYThe spurious pressures and ostensibly acceptable velocities which sometimes result from certain FEM approximate solutions of the incompressible Navier-Stokes equations are explained in detail. The concept of pressure modes, physical and spurious, pure and impure, is introduced and their effects on discretized solutions is analysed, in the context of both mixed interpolation and penalty approaches. Pressure filtering schemes, which are capable of recovering useful pressures from otherwise polluted numerical results, are developed for two particular elements in two-dimensions and one element in three-dimensions. The automatic pressure filter associated with the penalty method is also explained. Implications regarding the effect of spurious pressure modes on accuracy and ultimate convergence with mesh refinement are discussed and a list of unanswered questions presented. Sufficient numerical examples are discussed to corroborate the theory presented herein.

Low-order bubble-type stabilization

“…For the Franca element, the velocities and the pressure appear to be converging at the expected rate (O(h)). For this problem the apparent superconvergence (O(h 1:5 )) of the pressure for the mini element is in part due to the fact that all simulations were performed on structured meshes and also to the smoothness of the solution (see Reference [18]). …”

Review of literature on the finite-element solution of the equations of two-dimensional surface-water flow in the horizontal plane

“…By studying the behavior of the algebraic equations derived by applying the finite-element method to small patches of elements for steady Stokes flow, Sani, Gresho, Lee, and Griffiths ( 1981) and Sani, Gresho, Lee, Griffiths, and Engelman ( 1981) are able to characterize conditions giving rise to spurious pressure modes for a number of elements involving both equal-order and mixed interpolation. Since the pressure-mode problem for the steady Stokes equations is equivalent to the pressure-mode problem for the steady Navier-Stokes equations, and since the steady Navier-Stokes equations are equivalent to the steady shallow-water equations with pressure corresponding to depth and other appropriate identifications (Zienkiewicz and Heinrich, 1979, p. 681), the characterizations made by Sani and his coworkers apply equally well to the steady shallowwater equations.…”

“…Jackson and Cliffe (1981, p. 1663, 1675, who study mixed interpolation for the Navier-Stokes equations, note that specifying normal stresses instead of normal velocities on one boundary segment reduces the number of spurious pressure modes because the number of available test functions is increased. Sani, Gresho, Lee, and Griffiths (1981) and Sani, Gresho, Lee, Griffiths, and Engelman (1981) discuss the relationship between boundary conditions and the existence of spurious modes in various finite-element formulations of the Navier-Stokes equations. For example, it is shown that for mixed interpolation with continuous biquadratic velocity and discontinuous bilinear pressure (the pressure nodes are located at the 2 x 2 Gauss points), the single spurious pressure mode can be suppressed by avoiding the specification of tangential velocity components on the boundary (Sani, Gresho, Lee, Griffiths, and Engelman, 1981, p. 177).…”

Split, Characteristic Based Semi-Implicit Algorithm for Laminar/Turbulent Incompressible Flows

“…Step 4 Using 0 and P, evaluate K in the one equation model or both K and c in the two equation model…”