On an Uzawa smoother in multigrid for poroelasticity equations
On an Uzawa smoother in multigrid for poroelasticity equations
复制标题
多孔弹性方程多重网格 Uzawa 平滑器
DOI：
10.1002/nla.2074
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发表时间：
2017
影响因子：
4.3
通讯作者：
C. Oosterlee
中科院分区：
文献类型：
--
作者：
Peiyao Luo;C. Rodrigo;F. Gaspar;C. Oosterlee
多孔弹性饱和介质可以通过比奥固结理论来建模。它描述了多孔材料的变形与其内部流体流动之间随时间变化的相互作用。这里，为了有效求解孔隙弹性方程，采用多重网格方法，并使用 Uzawa 型迭代作为平滑器。 Uzawa 平滑器是一个方程程序。它应被解释为位移的对称高斯-赛德尔平滑与压力场中舒尔补的理查森迭代的组合。 Richardson迭代涉及影响收敛速度的松弛参数，必须仔细确定。平滑器的分析基于局部傅里叶分析（LFA）的框架，它使我们能够提供 Uzawa 平滑器的平滑因子的解析界限以及松弛参数的最佳值。数值实验表明，我们的上限提供了对精确平滑因子的令人满意的估计，并且所选的松弛参数是最优的。为了提高收敛性能，考虑了迭代重组加速多重网格。数值结果证实了加速方案的效率和鲁棒性。
A poroelastic saturated medium can be modeled by means of Biot's theory of consolidation. It describes the time‐dependent interaction between the deformation of porous material and the fluid flow inside of it. Here, for the efficient solution of the poroelastic equations, a multigrid method is employed with an Uzawa‐type iteration as the smoother. The Uzawa smoother is an equation‐wise procedure. It shall be interpreted as a combination of the symmetric Gauss‐Seidel smoothing for displacements, together with a Richardson iteration for the Schur complement in the pressure field. The Richardson iteration involves a relaxation parameter which affects the convergence speed, and has to be carefully determined. The analysis of the smoother is based on the framework of local Fourier analysis (LFA) and it allows us to provide an analytic bound of the smoothing factor of the Uzawa smoother as well as an optimal value of the relaxation parameter. Numerical experiments show that our upper bound provides a satisfactory estimate of the exact smoothing factor, and the selected relaxation parameter is optimal. In order to improve the convergence performance, the acceleration of multigrid by iterant recombination is taken into account. Numerical results confirm the efficiency and robustness of the acceleration scheme.