Numerical Analysis of Locally Adaptive Penalty Methods For The Navier-Stokes Equations

TL;DR

This work extends Xie's adaptive penalty approach from Stokes to the nonlinear, time-dependent Navier–Stokes equations, delivering unconditional stability, local control of the divergence, and a priori error estimates for a semi-discrete, locally adaptive penalty formulation. By updating the elementwise penalty parameter $\epsilon_{\Delta}$ according to a local tolerance $LocTol_{\Delta}$ via $\epsilon_{\Delta,\text{new}} = \rho\epsilon_{\Delta,\text{old}}$ with $\rho = LocTol_{\Delta}/\|\nabla\cdot u^h_\epsilon\|^2_{\Delta}$, the method keeps $\| abla\cdot u^h_\epsilon\|$ small while maintaining stability. Numerical tests on a modified Green–Taylor vortex and on flows between offset cylinders verify the predicted convergence and divergence control, and a third test demonstrates the benefits of combining the locally adaptive penalty with adaptive time stepping for flows with sharp transitions. The results suggest practical gains in robustness and efficiency for incompressible flow simulations and point to open questions on pressure recovery and conditioning in extended or coupled systems.

Abstract

Penalty methods relax the incompressibility condition and uncouple velocity and pressure. Experience with them indicates that the velocity error is sensitive to the choice of penalty parameter $ε$. So far, there is no effective á prior formula for $ε$. Recently, Xie developed an adaptive penalty scheme for the Stokes problem that picks the penalty parameter $ε$ self-adaptively element by element small where $\nabla \cdot u^h$ is large. Her numerical tests gave accurate fluid predictions. The next natural step, developed here, is to extend the algorithm with supporting analysis to the non-linear, time-dependent incompressible Navier-Stokes equations. In this report, we prove its unconditional stability, control of $\|\nabla \cdot u^h\|$, and provide error estimates. We confirm the predicted convergence rates with numerical tests.

Figures (7)

• Figure 1: Comparison of the divergence of velocity $\|\nabla \cdot u\|$ for adaptive $\epsilon$ penalty, constant penalty $\epsilon=\Delta t$, and non-penalized coupled NSE. The adaptive penalty method is far better.
• Figure 2: The $\epsilon_{min}$, $\epsilon_{max}$ and $\epsilon_{ave}$ among all the triangle meshes evolve for the adaptive $\epsilon$ penalty. After reaching a steady state, $\epsilon_{min}=EpsMin=10^{-10}$, $\epsilon_{max}=EpsMax=10^{-2}$ and $\epsilon_{ave} \approx 5 \times 10^{-4}$.
• Figure 3: Plot of $g(t)$.
• Figure 4: Timestep $\Delta t$ evolve in time.
• Figure 5: The local truncation error.

Theorems & Definitions (17)

• Lemma 2.1
• proof
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• Lemma 2.5
• Definition 2.1
• Lemma 2.6
• Theorem 3.1
• proof