Numerical Analysis of Locally Adaptive Penalty Methods For The Navier-Stokes Equations
<|reference_start|>Numerical Analysis of Locally Adaptive Penalty Methods For The Navier-Stokes Equations: 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 \'a prior formula for $\epsilon$. Recently, Xie developed an adaptive penalty scheme for the Stokes problem that picks the penalty parameter $\epsilon$ 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.<|reference_end|>
@article{fang2024numerical,
    title={Numerical Analysis of Locally Adaptive Penalty Methods For The
  Navier-Stokes Equations},
    author={Rui Fang},
    journal={arXiv preprint arXiv:2404.11712},
    year={2024},
    archivePrefix={arXiv},
    eprint={2404.11712},
    primaryClass={math.NA cs.NA}
}