论文标题
参数射击Braess-Sarazin型线性弹性问题的Smoother
Parameter-robust Braess-Sarazin-type smoothers for linear elasticity problems
论文作者
论文摘要
在这项工作中，我们提出了三个用于求解线性弹性问题的Braess-sarazin型多机松弛方案，其中标记和细胞方案是一种有限的差异方法，用于离散化。这三个放松方案是雅各比·布雷斯 - 萨拉赞，大众布雷斯 - 撒拉津和范卡·布雷斯 - 撒拉津。提出了针对块结构的松弛方案的局部傅立叶分析（LFA），以研究多族收敛行为。从LFA，我们为每种情况得出了最佳的LFA平滑因子。我们获得高效的平滑因子，这些因子与拉梅常数无关。 Vanka-Braess-sarazin松弛方案导致最有效的方案。在每个放松中，需要解决Schur补体系统。由于直接求解通常很昂贵，因此开发了一个不精确的版本，我们只能在Schur补体系统上最多使用三个加权雅各比迭代。最后，提出了两个网格和V周期的多移民表演，以验证我们的理论结果。我们的数值结果表明，不确定的版本可以实现与精确版本相同的性能，而我们的方法对Lamé常数具有鲁棒性。
In this work, we propose three Braess-Sarazin-type multigrid relaxation schemes for solving linear elasticity problems, where the marker and cell scheme, a finite difference method, is used for the discretization. The three relaxation schemes are Jacobi-Braess-Sarazin, Mass-Braess-Sarazin, and Vanka-Braess-Sarazin. A local Fourier analysis (LFA) for the block-structured relaxation schemes is presented to study multigrid convergence behavior. From LFA, we derive optimal LFA smoothing factor for each case. We obtain highly efficient smoothing factors, which are independent of Lamé constants. Vanka-Braess-Sarazin relaxation scheme leads to the most efficient one. In each relaxation, a Schur complement system needs to be solved. Due to the fact that direct solve is often expensive, an inexact version is developed, where we simply use at most three weighted Jacobi iterations on the Schur complement system. Finally, two-grid and V-cycle multigrid performances are presented to validate our theoretical results. Our numerical results show that inexact versions can achieve the same performance as that of exact versions and our methods are robust to the Lamé constants.
