A pressure-robust discretization of Oseen's equation using stabilization in the vorticity equation
A pressure-robust discretization of Oseen's equation using stabilization in the vorticity equation
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DOI：
10.1137/20m1351230
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发表时间：
2020-07
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影响因子：
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通讯作者：
N. Ahmed;G. Barrenechea;E. Burman;Johnny Guzm'an;A. Linke;C. Merdon
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文献类型：
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作者：
N. Ahmed;G. Barrenechea;E. Burman;Johnny Guzm'an;A. Linke;C. Merdon
对于高雷诺数状态，考虑使用压力鲁棒有限元方法对纳维-斯托克斯方程进行离散化。为了对抗由于对流占主导地位的振荡，我们添加了基于体项的稳定，其形式是涡量方程的基于残差的最小二乘稳定，并辅以元件表面上梯度跳跃（的某些分量）的惩罚项。由于稳定是基于涡量方程，因此它与压力梯度无关，这使得它具有抗压能力。因此，我们证明了线性化情况下与压力无关的误差估计，称为奥辛问题。事实上，我们证明了 $L^2$-范数中的 $O(h^{k+\frac12})$ 误差估计，已知这是此类问题的最佳估计。提供的数值例子除了证实了理论结果之外，还表明本方法优于经典的基于残差的 SUPG 稳定方法。
Discretization of Navier-Stokes' equations using pressure-robust finite element methods is considered for the high Reynolds number regime. To counter oscillations due to dominating convection we add a stabilization based on a bulk term in the form of a residual-based least squares stabilization of the vorticity equation supplemented by a penalty term on (certain components of) the gradient jump over the elements faces. Since the stabilization is based on the vorticity equation, it is independent of the pressure gradients, which makes it pressure-robust. Thus, we prove pressure-independent error estimates in the linearized case, known as Oseen's problem. In fact, we prove an $O(h^{k+\frac12})$ error estimate in the $L^2$-norm that is known to be the best that can be expected for this type of problem. Numerical examples are provided that, in addition to confirming the theoretical results, show that the present method compares favorably to the classical residual-based SUPG stabilization.