On the Divergence Constraint in Mixed Finite Element Methods for Incompressible Flows

Abstract

The divergence constraint of the incompressible Navier-Stokes equations is revisited in the mixed finite element framework. While many stable and convergent mixed elements have been developed throughout the past four decades, most classical methods relax the divergence constraint and only enforce the condition discretely. As a result, these methods introduce a pressure-dependent consistency error which potentially might pollute the computed velocity. Mathematically, these methods are not robust in the sense that a contribution from the righthand side which influences only the pressure in the continuous equations possesses an impact on both velocity and pressure in the discrete equations. This paper reviews the theory and practical implications of relaxing the divergence constraint. Several approaches for improving the discrete mass balance or even for computing divergencefree solutions will be discussed: grad-div stabilization, higher order mixed methods derived on the basis of an exact de Rham complex, H(div)-conforming finite elements, and mixed methods which an appropriate projection of the test function. Numerical examples illustrate both the potential effects of using non-robust discretizations and the improvements obtained with utilizing robust discretizations.

A locally conservative and energy‐stable finite‐element method for the Navier‐Stokes problem on time‐dependent domains

“…Consider Equation (20) in the first space-time slab  n , n = 0. Assume that we are given a solution (20),…”

“…17 Using space-time hexahedral cells, they were able to obtain optimal rates of convergence for the velocity, velocity gradient, and pressure fields on time-dependent domains. 20 This lack of 'pressure-robustness' may cause large errors in the velocity, especially for convection dominated flows. 19 The absence of a velocity field that is pointwise divergence-free and H(div)-conforming also has other consequences, for example, the velocity error will depend on the pressure error scaled by the inverse of the viscosity.…”

Virtual Elements for the Navier--Stokes Problem on Polygonal Meshes

“…In the survey work [39], the authors classify the divergence-free mixed finite element methods into three groups: (1) conforming schemes, (2) discontinuous Galerkin schemes, and (3) schemes with an appropriate reconstruction of the test functions.…”

“…We are now ready to state the proposed discrete problem. Referring to (20), (21), (29), (39), and (23), we consider the virtual element problem:…”

“…These, in turn, suggest making use of an enhanced discrete velocity space [52], which is an improvement with respect We finally note that, currently, there do exist Galerkin-type finite element methods for the Stokes and Navier--Stokes equations that are pressure-robust (that is, the error on the velocity does not depend on the pressure, not even indirectly through the loading or convection terms). Some recent examples of such schemes can be found in [40,34], while a comprehensive review is provided in [39]. However, to the best of our knowledge, all of the available schemes work only for standard simplicial/hexahedral meshes.…”