A Note on Accurate and Efficient Higher Order Galerkin Time Stepping Schemes for the Nonstationary Stokes Equations
A Note on Accurate and Efficient Higher Order Galerkin Time Stepping Schemes for the Nonstationary Stokes Equations
复制标题
非平稳斯托克斯方程准确高效的高阶伽辽金时间步进方案的注解
DOI：
10.2174/1876389801204010035
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发表时间：
2012
期刊：
影响因子：
--
通讯作者：
S. Turek
中科院分区：
文献类型：
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作者：
S. Hussain;F. Schieweck;S. Turek
在这篇文章中，我们扩展了我们最近在[1]中关于热方程的工作，并通过数值实验描述和比较了连续Galerkin-Petrov（cGP）和不连续Galerkin（dG）时间离散应用于二维情况下的非平稳Stokes方程。对于空间离散化，我们使用著名的 LBB 稳定四边形有限元，它由速度的一致双二次元和压力的不连续线性元组成。我们讨论了实现方面以及使用基于 Vanka 型平滑器的整体多重网格求解器求解所得块系统的方法。通过数值实验，我们比较了不同时间离散化的准确性和计算成本。我们表明，多重网格方法的收敛行为几乎独立于空间中的网格大小和时间步长，这意味着（至少对于这些示例）我们已经创建了一个有效的求解过程。 2000 年数学科目分类 (MSC)：65M12、65M55、65M60。
In this note, we extend our recent work for the heat equation in [1] and describe and compare by means of numerical experiments the continuous Galerkin-Petrov (cGP) and discontinuous Galerkin (dG) time discretization applied to the nonstationary Stokes equations in the two-dimensional case. For the space discretization, we use the well-known LBB-stable quadrilateral finite element which consists of conforming biquadratic elements for the velocity and discontinuous linear elements for the pressure. We discuss implementation aspects as well as methods for solving the resulting block systems using monolithic multigrid solvers based on Vanka-type smoothers. By means of numerical experiments we compare the different time discretizations with respect to accuracy and computational costs. We show that the convergence behavior of the multigrid method is almost independent of the mesh size in space and the time step size which means (at least for these examples) that we have created an efficient solution process. 2000 Mathematics Subject Classification (MSC): 65M12, 65M55, 65M60.