This article presents an efficient stabilized finite element scheme for solving viscoelastic flow problems at high Weissenberg numbers. The velocity and pressure variables in the momentum balance equations are uncoupled using an incremental fractional step method based on the second‐order backward differentiation formula. The pressure gradient projection stabilization technique and the discrete elastic‐viscous‐split‐stress formulation are introduced into the scheme, in explicit versions, to circumvent the LBB constraints. For the constitutive equation, a square root transformation is first applied to preserve the positive definiteness of the polymer conformation tensor, and then the streamline upwind/Petrov–Galerkin method and the second‐order Runge–Kutta scheme are implemented for spatial and time discretizations. Three benchmark problems are tested and numerical results have revealed the accuracy and convergence of the scheme. What is more, since the presented scheme enables the use of equal low‐order interpolations for all variables, and requires no iterative process, it is computationally efficient and easy to be implemented.

中文翻译：

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一种模拟粘弹性流的有效稳定有限元方案

本文提出了一种有效的稳定有限元方案，用于解决高魏森伯格数下的粘弹性流动问题。动量平衡方程中的速度和压力变量使用基于二阶后向微分公式的增量分数步法解耦。该方案采用显式形式引入了压力梯度投影稳定技术和离散的弹性-粘滞-分离-应力公式，以规避LBB约束。对于本构方程，首先应用平方根变换来保留聚合物构象张量的正定性，然后对空间和时间离散化采用流线上风/ Petrov-Galerkin方法和二阶Runge-Kutta方案。测试了三个基准问题，数值结果表明了该方案的准确性和收敛性。不仅如此，由于所提出的方案可以对所有变量使用相等的低阶插值，并且不需要迭代过程，因此它的计算效率很高，易于实现。