non-stationary incompressible Navier-Stokes equations
Glowinski's fractional step theta-scheme
Parallel solution of the instationary incompressible Navier-Stokes equations
This work is concerned with combination of multigrid and parallelization techniques for solving the non-stationary, incompressible Navier-Stokes problem, especially on anisotropic meshes. The main focus is on aspects of implementation and efficiency. NEWLINENEWLINENEWLINEThe Navier-Stokes equations are firstly discretized in space by a non-conforming finite element method that is based upon a triangulation into hexahedrons, rotated trilinear ansatz functions for the velocity, and a piecewise constant pressure approximation. For stabilization, simple and Samarskij upwinding are employed. The time discretization relies upon Glowinski's fractional step \(\theta\)-scheme, and the velocity and pressure problem are decoupled by van Kan's projection method. The nonlinear problems are solved with a fixed-point defect-correction method. A detailed analysis of the spatial discretization can be found in [\textit{F. Schieweck}, Parallele Lösung der stationären inkompressiblen Navier-Stokes Gleichungen. (Parallel solution of the stationary incompressible Navier-Stokes equations). (German) Magdeburg: Univ. Magdeburg, Fakultät Mathematik (1997; Zbl 0915.76051)]. NEWLINENEWLINENEWLINEThe parallelization is based upon a partition of the coarse grid. The resulting parallel blocks are then uniformly refined. The author compares the suggested parallel multigrid method with its sequential counterpart; they differ only in the smoothing process. NEWLINENEWLINENEWLINEIt turns out that the parallel method is essentially of the same numerical efficiency as the corresponding sequential method. However, the choice of the coarse grid (in particular, the relation of the numbers of coarse grid elements and processors) strongly influences the parallel efficiency. The best results are obtained for the parallel CG-method with multiplicative multigrid for the preconditioning. NEWLINENEWLINENEWLINEThe method suggested is tested on lid-driven cavity problem with a Reynolds number \(Re = 3200\), and on three-dimensional flow around a cylinder with \(Re = 100\) (with \(48\times 10^6\) unknowns on a Cray T3E/512). Finally, the author provides the C++ class library ParFlow++ that allows to develop parallel solvers for incompressible flow problems.
scientific article; zbMATH DE number 1671895