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 \(\epsilon \) . So far, there is no effective á priori formula for \(\epsilon \) . Recently, Xie developed an adaptive penalty scheme for the Stokes problem that picks the penalty parameter \(\epsilon \) self-adaptively mesh element by mesh 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 \(\Vert \nabla \cdot u^h\Vert \) , and provide error estimates. We confirm the predicted convergence rates with numerical tests.
