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

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.