Journal of Computational Science ( IF 3.3 ) Pub Date : 2021-01-20 , DOI: 10.1016/j.jocs.2021.101306 Marcin Łoś , Sergio Rojas , Maciej Paszyński , Ignacio Muga , Victor M. Calo
In this paper, we introduce a stable isogeometric analysis discretization of the Stokes system of equations. We use this standard constrained problem to demonstrate the flexibility and robustness of the residual minimization method on dual stable norms [16], which unlocks the extraordinary approximation power of isogeometric analysis [44]. That is, we introduce an isogeometric residual minimization method (IRM) for the Stokes equations, which minimizes the residual in a dual discontinuous Galerkin norm; thus we use the acronym DGiRM. Following Calo et al. [16], we start from an inf-sup stable discontinuous Galerkin (DG) formulation to approximate in a highly continuous trial space that minimizes the dual norm of the residual in a discontinuous test space. We demonstrate the performance and robustness of the methodology considering a manufactured solution and the well-known lid-driven cavity flow problem. First, we use a multi-frontal direct solver, and, using the Pareto front, compare the resulting numerical accuracy and the computational cost expressed by the number of floating-point operations performed by the direct solver algorithm. Second, we use an iterative solver. We measure the number of iterations required when increasing the mesh size and how the configuration of spaces affect the resulting accuracy. This paper is an extension of the paper A Stable Discontinuous Galerkin Based Isogeometric Residual Minimization for the Stokes Problem by M. Łoś, et al. (2020) published in Lecture Notes in Computer Science. In this extended version, we deepen in the mathematical aspects of the DGiRM, include the iterative solver algorithm and implementation, and discuss the influence of different discretization spaces on the iterative solver's convergence.
中文翻译:
DGIRM:基于不连续Galerkin的Stokes问题等几何残差最小化
在本文中,我们介绍了Stokes方程组的稳定等几何分析离散化。我们使用这个标准约束问题来证明对偶稳定范数的残差最小化方法的灵活性和鲁棒性[16],从而释放了等几何分析的非凡逼近能力[44]。也就是说,我们为Stokes方程引入了等几何残差最小化方法(IRM),该方法将对偶不连续Galerkin范数中的残差最小化。因此,我们使用缩写DGiRM。继卡洛等。[16],我们从一个无限稳定的不连续Galerkin(DG)公式开始,在一个高度连续的试验空间中进行近似,从而最大程度地减少了在不连续试验空间中残差的对偶范数。我们展示了一种方法的性能和鲁棒性,考虑了制造解决方案和众所周知的盖驱动腔流动问题。首先,我们使用多面直接求解器,然后使用Pareto前沿比较所得的数值精度和由直接求解器算法执行的浮点运算次数表示的计算成本。其次,我们使用迭代求解器。我们测量增加网格大小时所需的迭代次数,以及空间的配置如何影响最终的准确性。本文是本文的扩展 比较所得的数值精度和由直接求解器算法执行的浮点运算数量表示的计算成本。其次,我们使用迭代求解器。我们测量增加网格大小时所需的迭代次数,以及空间的配置如何影响最终的准确性。本文是本文的扩展 比较所得的数值精度和由直接求解器算法执行的浮点运算数量表示的计算成本。其次,我们使用迭代求解器。我们测量增加网格大小时所需的迭代次数,以及空间的配置如何影响最终的准确性。本文是本文的扩展M.Łoś等人针对Stokes问题基于稳定不连续Galerkin的等几何残差最小化。(2020年)发表在《计算机科学讲座笔记》中。在此扩展版本中,我们加深了DGiRM的数学方面的内容,包括迭代求解器算法和实现,并讨论了不同离散空间对迭代求解器收敛的影响。