Pressure robust SUPG-stabilized finite elements for the unsteady Navier-Stokes equation

TL;DR

The paper introduces a conforming finite element method for the unsteady Navier–Stokes equations that is both convection quasi-robust and pressure robust by marrying a divergence-free velocity–pressure pair (such as Scott–Vogelius), a discontinuous Galerkin discretization in time, and a curl-based SUPG stabilization of the vorticity. Theoretical analysis proves stability and convergence under standard mesh and regularity assumptions, with velocity errors that are independent of the diffusion coefficient $\nu$ and exhibit an enhanced convergence rate $O(h^{k+1/2})$ in convection-dominated regimes. The method preserves the divergence-free constraint at the discrete level and demonstrates pressure robustness, with velocity errors unaffected by pressure changes and resilient to small viscosities. Numerical experiments corroborate the theory, showing optimal convergence in both regimes, pressure-robust velocity solutions at all viscosities, and superior performance over non-stabilized schemes in lattice vortex dynamics.

Abstract

In the present contribution we propose a novel conforming Finite Element scheme for the time-dependent Navier-Stokes equation, which is proven to be both convection quasi-robust and pressure robust. The method is built combining a "divergence-free" velocity/pressure couple (such as the Scott-Vogelius element), a Discontinuous Galerkin in time approximation, and a suitable SUPG-curl stabilization. A set of numerical tests, in accordance with the theoretical results, is included.

Figures (9)

• Figure 1: Barycenter-refined mesh 2 (left). Mesh size $h$ and associated time step size $\tau$ for all refinement levels (right).
• Figure 2: Convergence analysis, $\nu = \texttt{1}$ (diffusion dominated regime). Convergence histories of both $\text{err}(\boldsymbol{U}, H^1)$ and $\text{err}(\boldsymbol{U}, L^2)$, left, and $\text{err}(P, L^2)$ right.
• Figure 3: Convergence analysis, $\nu = \texttt{1e-11}$ (diffusion dominated regime). Convergence histories of $\text{err}(\boldsymbol{U}, L^2)$ and $\text{err}(P, L^2)$.
• Figure 4: Convergence analysis: errors $\text{err}(\boldsymbol{U}, L^2)$ and $\text{err}(P, L^2)$ as a function of the viscosity $\nu$ on mesh 3.
• Figure 5: Convergence analysis: sensitivity analysis of the parameter $c_E$ on $\text{err}(\boldsymbol{U}, L^2)$ and $\text{err}(P, L^2)$ for $\nu = \texttt{1e-11}$ and $\nu = \texttt{1}$ considering mesh 2.

Theorems & Definitions (37)

• Remark 1: Generalized notation
• Lemma 3.1: Stream functions
• Lemma 3.2: Trace inequality
• Lemma 3.3: Bramble-Hilbert
• Lemma 3.4: Inverse estimate
• Remark 2
• Lemma 3.5: Time-interpolation
• Proposition 3.1: Gronwall lemma
• Lemma 4.1
• proof