论文标题
当地Vanka的效率平滑的几何多物种元素预处理，用于时空有限元方法到Navier-Stokes方程
Efficiency of local Vanka smoother geometric multigrid preconditioning for space-time finite element methods to the Navier-Stokes equations
论文作者
论文摘要
不可压缩的粘性流的数值模拟，尤其是在三个空间维度中，仍然仍然是一项具有挑战性的任务。时空有限元方法具有自然构建高阶离散化方案。他们提供了在计算可行的网格上获得准确结果的潜力。通过牛顿方法线性化产生的代数问题可产生用$（k+1）\ times（k+1）$ saddle点系统构建的块矩阵的线性系统，其中$ k $表示各种时间离散化的多项式顺序。我们从数字上证明了通过$ V $ cycle-Cypotric Multigrid方法基于局部Vanka更平滑的$ V $ Cypopric Multigrid方法来解决这些线性系统的预处理效率。这些研究是针对围绕圆柱体流动的二维基准问题进行的。在这里，分析并在数值上分析了求解器相对于分段多项式$ k $的鲁棒性。
Numerical simulation of incompressible viscous flow, in particular in three space dimensions, continues to remain a challenging task. Space-time finite element methods feature the natural construction of higher order discretization schemes. They offer the potential to achieve accurate results on computationally feasible grids. Linearizing the resulting algebraic problems by Newton's method yields linear systems with block matrices built of $(k+1)\times (k+1)$ saddle point systems, where $k$ denotes the polynomial order of the variational time discretization. We demonstrate numerically the efficiency of preconditioning GMRES iterations for solving these linear systems by a $V$-cycle geometric multigrid approach based on a local Vanka smoother. The studies are done for the two- and three-dimensional benchmark problem of flow around a cylinder. Here, the robustness of the solver with respect to the piecewise polynomial order $k$ in time is analyzed and proved numerically.
