Stabilizing poor mass conservation in incompressible flow problems with large irrotational forcing and application to thermal convection

We consider the problem of poor mass conservation in mixed finite element algorithms for flow problems with large rotation-free forcing in the momentum equation. We provide analysis that suggests for such problems, obtaining accurate solutions necessitates either the use of pointwise divergence-free finite elements (such as Scott–Vogelius), or heavy grad-div stabilization of weakly divergence-free elements. The theory is demonstrated in numerical experiments for a benchmark natural convection problem, where large irrotational forcing occurs with high Rayleigh numbers.

► If irrotational part of forcing is large, then O(1) grad-div stabilization in mixed FEM for NSE is likely not optimal.
► In such cases, our analysis reveals that heavy grad-div stabilization or pointwise div-free mixed FEM is better.
► We give a physically relevant example: benchmark natural convection with large Rayleigh number.
► We test Scott–Vogelius, Taylor–Hood, (P2, P0), and (P1Bub, P1) elements, with grad-div stabilization.
► Here we see good solutions only if the divergence is sufficiently small, thus validating our theory.