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.
翻译:特别是在三个空间维度上,不可压缩的透视流的数值模拟仍然是一项具有挑战性的任务。空间-时间有限元素方法是高顺序离散计划的自然构造特征。它们提供了在计算可行的网格上取得准确结果的潜力。牛顿方法对由此产生的代数问题进行线性分析后产生线性系统,其块状矩阵以$(k+1)\time(k+1)=timlepole point systems(k+1)=xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx