Due to simplicity in implementation and data structure, elements with equal‐order interpolation of velocity and pressure are very popular in finite‐element‐based flow simulations. Although such pairs are inf‐sup unstable, various stabilization techniques exist to circumvent that and yield accurate approximations. The most popular one is the pressure‐stabilized Petrov–Galerkin (PSPG) method, which consists of relaxing the incompressibility constraint with a weighted residual of the momentum equation. Yet, PSPG can perform poorly for low‐order elements in diffusion‐dominated flows, since first‐order polynomial spaces are unable to approximate the second‐order derivatives required for evaluating the viscous part of the stabilization term. Alternative techniques normally require additional projections or unconventional data structures. In this context, we present a novel technique that rewrites the second‐order viscous term as a first‐order boundary term, thereby allowing the complete computation of the residual even for lowest‐order elements. Our method has a similar structure to standard residual‐based formulations, but the stabilization term is computed globally instead of only in element interiors. This results in a scheme that does not relax incompressibility, thereby leading to improved approximations. The new method is simple to implement and accurate for a wide range of stabilization parameters, which is confirmed by various numerical examples.

中文翻译：

---

不可压缩流的等阶有限元逼近的基于全局残差的镇定

由于实现和数据结构的简化，具有速度和压力的等阶插值的元素在基于有限元的流模拟中非常受欢迎。尽管这样的对不稳定，但仍存在各种稳定技术来规避这种情况并产生精确的近似值。最受欢迎的一种方法是压力稳定的Petrov-Galerkin（PSPG）方法，该方法包括用动量方程的加权残差放宽不可压缩性约束。然而，由于一阶多项式空间无法近似评估稳定项粘性部分所需的二阶导数，因此PSPG对于扩散为主的流中的低阶元素可能表现不佳。替代技术通常需要其他预测或非常规数据结构。在这种情况下，我们提出了一种将二阶粘性项重写为一阶边界项的新颖技术，从而即使对于最低阶元素也可以完整计算残差。我们的方法与基于标准残差的公式具有相似的结构，但是稳定项是全局计算的，而不是仅在元素内部计算。这导致不放松不可压缩性的方案，从而导致改善的近似性。该新方法易于实现，并且对于各种稳定参数都具有精确性，这一点已通过各种数值示例得到了证实。我们的方法与基于标准残差的公式具有相似的结构，但是稳定项是全局计算的，而不是仅在元素内部计算。这导致不放松不可压缩性的方案，从而导致改善的近似性。该新方法易于实现，并且对于各种稳定参数都具有精确性，这一点已通过各种数值示例得到了证实。我们的方法与基于标准残差的公式具有相似的结构，但是稳定项是全局计算的，而不是仅在元素内部计算。这导致不放松不可压缩性的方案，从而导致改善的近似性。该新方法易于实现，并且对于各种稳定参数都具有精确性，这一点已通过各种数值示例得到了证实。