A quantitative performance study for Stokes solvers at the extreme scale

• Published in: Journal of computational science Vol. 17; pp. 509 - 521
• Main Authors: Gmeiner, Björn, Huber, Markus, John, Lorenz, Rüde, Ulrich, Wohlmuth, Barbara
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01.11.2016
• Subjects: Flow around sphere pack; Hierarchical hybrid grids; Parallel textbook multigrid efficiency; Scalability; Stokes system; Uzawa smoother; Hierarchical hybrid grids; 68Q25; Stokes system; Scalability; Parallel textbook multigrid efficiency; Uzawa smoother; Flow around sphere pack; 65N55; 65Y05
• ISSN: 1877-7503; 1877-7511
• DOI: 10.1016/j.jocs.2016.06.006

•A quantitative performance study for Stokes solvers at extreme scales is presented.•Indefinite systems with up to 1013 unknowns and 786432 parallel threads are solved.•Parallel textbook multigrid efficiency is extended to a all-at-once multigrid method.•Simulation results of fluid flow in a pipe filled with spherical obstacles are shown. This article presents a systematic quantitative performance study for large finite element computations on extreme scale computing systems. Three parallel iterative solvers for the Stokes system, discretized by low order tetrahedral elements, are compared with respect to their numerical efficiency and their scalability running on up to 786432 parallel threads. An all-at-once multigrid method for the saddle point system using an Uzawa-type smoother provides the best overall performance with respect to memory consumption and time-to-solution. The largest system solved on a Blue Gene/Q system has more than ten trillion (1.1×1013) unknowns and requires about 13min compute time. Despite the matrix free and highly optimized implementation, the memory requirement for the solution vector and the auxiliary vectors is about 200TByte. A generalization of Brandt's notion of “textbook multigrid efficiency” is employed to study the algorithmic performance of the all-at-once multigrid solver at the extreme scale. The flexibility of the method is demonstrated for simulating incompressible fluid flow in a pipe filled with spherical obstacles.