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

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.