Performance and Scalability of Hierarchical Hybrid Multigrid Solvers for Stokes Systems

In many applications involving incompressible fluid flow, the Stokes system plays an important role. Complex flow problems may require extremely fine resolutions, easily resulting in saddle-point problems with more than a trillion ($10^{12}$) unknowns. Even on the most advanced supercomputers, the fast solution of such systems of equations is a highly nontrivial and challenging task. In this work we consider a realization of an iterative saddle-point solver which is based mathematically on the Schur-complement formulation of the pressure and algorithmically on the abstract concept of hierarchical hybrid grids. The design of our fast multigrid solver is guided by an innovative performance analysis for the computational kernels in combination with a quantification of the communication overhead. Excellent node performance and good scalability to almost a million parallel threads are demonstrated on different characteristic types of modern supercomputers.