Weighted BFBT Preconditioner for Stokes Flow Problems with Highly Heterogeneous Viscosity
Weighted BFBT Preconditioner for Stokes Flow Problems with Highly Heterogeneous Viscosity
复制标题
用于解决高非均质粘度斯托克斯流问题的加权 BFBT 预处理器
DOI：
10.1137/16m108450x
复制
发表时间：
2016
期刊：
影响因子：
--
通讯作者：
O. Ghattas
中科院分区：
文献类型：
--
作者：
J. Rudi;G. Stadler;O. Ghattas
本文提出了一种加权的BFBT近似（w-BFBT）来近似具有高度非均匀粘性的Stokes系统的逆Schur补。当作为一个舒尔互补为基础的斯托克斯预条件的一部分，我们观察到强大的快速收敛的斯托克斯问题光滑，但高度变化（高达10个数量级）的粘度，最佳算法的可扩展性方面的网格细化，只有轻微的依赖于高阶有限元离散的多项式阶（$Q_k \times P_{k-1}^{disc}$，阶$k \ge 2$）。对于某些困难的问题，我们证明了数值w-BFBT显着提高斯托克斯求解器的收敛性，广泛使用的逆粘度加权压力质量矩阵近似的舒尔补。此外，我们推导出理论上的特征值界，以证明谱等价的w-BFBT。使用详细的数值实验，我们讨论修改w-BFBT在Dirichlet边界，减少迭代次数。斯托克斯求解器的整体算法性能是由功效的w-BFBT作为舒尔补近似，此外，由我们的并行混合谱几何代数多重网格（HMG）方法，我们使用近似逆的粘性块和变系数压力泊松算子内w-BFBT。基于HMG的可扩展性，我们的Stokes求解器实现了90%的并行效率，同时从TACC的Lonestar 5超级计算机的48个核心到所有30，000个核心增加了600多倍。
We present a weighted BFBT approximation (w-BFBT) to the inverse Schur complement of a Stokes system with highly heterogeneous viscosity. When used as part of a Schur complement-based Stokes preconditioner, we observe robust fast convergence for Stokes problems with smooth but highly varying (up to 10 orders of magnitude) viscosities, optimal algorithmic scalability with respect to mesh refinement, and only a mild dependence on the polynomial order of high-order finite element discretizations ($Q_k \times P_{k-1}^{disc}$, order $k \ge 2$). For certain difficult problems, we demonstrate numerically that w-BFBT significantly improves Stokes solver convergence over the widely used inverse viscosity-weighted pressure mass matrix approximation of the Schur complement. In addition, we derive theoretical eigenvalue bounds to prove spectral equivalence of w-BFBT. Using detailed numerical experiments, we discuss modifications to w-BFBT at Dirichlet boundaries that decrease the number of iterations. The overall algorithmic performance of the Stokes solver is governed by the efficacy of w-BFBT as a Schur complement approximation and, in addition, by our parallel hybrid spectral-geometric-algebraic multigrid (HMG) method, which we use to approximate the inverses of the viscous block and variable-coefficient pressure Poisson operators within w-BFBT. Building on the scalability of HMG, our Stokes solver achieves a parallel efficiency of 90% while weak scaling over a more than 600-fold increase from 48 to all 30,000 cores of TACC's Lonestar 5 supercomputer.