On the Divergence Constraint in Mixed Finite Element Methods for Incompressible Flows
On the Divergence Constraint in Mixed Finite Element Methods for Incompressible Flows
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DOI：
10.1137/15m1047696
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发表时间：
2017-09-01
期刊：
影响因子：
10.2
通讯作者：
Rebholz, Leo G.
中科院分区：
文献类型：
--
作者：
John, Volker;Linke, Alexander;Rebholz, Leo G.
在混合有限元框架下重新讨论了不可压缩Navier-Stokes方程的发散约束。虽然在过去的四十年中已经开发了许多稳定和收敛的混合元，但大多数经典方法放松了发散约束，只离散地强制执行条件。因此，这些方法引入了压力相关的一致性误差，这可能会污染计算的速度。这些方法并不稳健，因为来自右侧的贡献仅影响连续方程中的压力，而影响离散方程中的速度和压力。本文综述了放松发散约束的理论和实践意义。几种方法，以提高离散的质量平衡，甚至计算发散的解决方案将被讨论：梯度div稳定，高阶混合方法的基础上得出的一个确切的德拉姆复杂，H（div）-符合有限元，混合方法与适当的重建的测试功能。数值例子说明了使用非鲁棒离散化的潜在影响和通过利用压力鲁棒离散化获得的改进。
The divergence constraint of the incompressible Navier-Stokes equations is revisited in the mixed finite element framework. While many stable and convergent mixed elements have been developed throughout the past four decades, most classical methods relax the divergence constraint and only enforce the condition discretely. As a result, these methods introduce a pressure-dependent consistency error which can potentially pollute the computed velocity. These methods are not robust in the sense that a contribution from the right-hand side, which influences only the pressure in the continuous equations, impacts both velocity and pressure in the discrete equations. This article reviews the theory and practical implications of relaxing the divergence constraint. Several approaches for improving the discrete mass balance or even for computing divergence-free solutions will be discussed: grad-div stabilization, higher order mixed methods derived on the basis of an exact de Rham complex, H(div)-conforming finite elements, and mixed methods with an appropriate reconstruction of the test functions. Numerical examples illustrate both the potential effects of using nonrobust discretizations and the improvements obtained by utilizing pressure-robust discretizations.