SIAM Journal on Scientific Computing, Volume 43, Issue 2, Page A1336-A1361, January 2021.
We propose a class of adaptive enriched Galerkin-characteristics finite element methods for efficient numerical solution of the incompressible Navier--Stokes equations in primitive variables. The proposed approach combines the modified method of characteristics to deal with convection terms, the finite element discretization to manage irregular geometries, a direct conjugate gradient algorithm to solve the Stokes problem, and an adaptive ${L}^2$-projection using quadrature rules to improve the efficiency and accuracy of the method. In the present study, the gradient of the velocity field is used as an error indicator for adaptation of enrichments by increasing the number of quadrature points where it is needed without refining the mesh. Unlike other adaptive finite element methods for incompressible Navier--Stokes equations, linear systems in the proposed enriched Galerkin-characteristics finite element method preserve the same structure and size at each refinement in the adaptation procedure. We examine the performance of the proposed method for a coupled Burgers problem with known analytical solution and for the benchmark problem of flow past a circular cylinder. We also solve a transport problem in the Mediterranean Sea to demonstrate the ability of the method to resolve complex flow features in irregular geometries. Comparisons to the conventional Galerkin-characteristics finite element method are also carried out in the current work. The computed results support our expectations for an accurate and highly efficient enriched Galerkin-characteristics finite element method for incompressible Navier--Stokes equations.

中文翻译：

---

不可压缩Navier-Stokes方程的丰富Galerkin特征有限元方法

SIAM科学计算杂志，第43卷，第2期，第A1336-A1361页，2021年1月。
针对原始变量中不可压缩的Navier-Stokes方程的有效数值解，我们提出了一类自适应丰富的Galerkin特征有限元方法。所提出的方法结合了处理对流项的改进特征方法，处理不规则几何的有限元离散化，解决斯托克斯问题的直接共轭梯度算法以及使用正交规则的自适应$ {L} ^ 2 $投影以提高该方法的效率和准确性。在本研究中，速度场的梯度通过增加需要正交点的数量而无需细化网格而被用作自适应富集的误差指标。与其他不可压缩的Navier-Stokes方程的自适应有限元方法不同，所提出的丰富的Galerkin特征有限元方法中的线性系统在自适应过程中的每次细化时都保持相同的结构和大小。我们用已知的解析解检查了耦合Burgers问题和流过圆柱的基准问题的方法的性能。我们还解决了地中海的运输问题，以证明该方法能够解决不规则几何形状中的复杂流动特征。在当前的工作中，还与传统的Galerkin特征有限元方法进行了比较。计算结果支持了我们对于不可压缩的Navier-Stokes方程的准确和高效的丰富Galerkin特征有限元方法的期望。