Experiences with Robust Parallel Multiple Semicoarsened Grid Methods

T. Washio ^{∗} and C.W. Oosterlee ^{‡}

Abstract:

In this paper robust parallel multilevel methods based on a nonstandard grid sequence are investigated for 2D and 3D scalar equations. The problems include the rotated anisotropic diffusion equation, Poisson type equations on stretched grids, Poisson problems with discontinuous coefficients, and convection-diffusion equations. The multilevel methods are used as a solver or as a preconditioner for Krylov subspace methods. The parallel performance of these methods is our main interest.

For 2D problems, the method is based on standard coarsening. However, semicoarsening MG V-cycles along the horizontal and vertical direction are employed sequentially as smoother on each grid. Hence we can attain robustness for any type of anisotropy with point Jacobi or Gauss Seidel relaxation on each grid. This technique is called the MultiGrid as Smoother (MG-S) method in [3]. In this paper, we extend the technique for 3D problems. Here, we apply the MG-S method in two directions (Y- and Z-direction), and use a parallel line Jacobi or Gauss-Seidel relaxation along the third direction (X). So the basic cycle is along the grid sequence in the Y- and Z-direction. The semicoarsening MG V-cycles along the Y- and Z-direction are applied sequentially as smoother with line relaxation in X-direction on each grid. As a result, the parallel solver is robust and applicable for strong coupling of unknowns along any direction.

The 2D and 3D nonstandard multilevel methods employ the Galerkin approach for constructing matrices on the (multiple coarse) grids, where the grid transfer operators and the matrices are automatically defined from the fine grid matrix [1], [4]. The technique to split the rectangular computational domain into different blocks is grid partitioning and communication is performed by means of message-passing on all grids. The comparison in this paper includes the performances of several matrix transfer operators, of two MG cycles (V and F) on the basic grid sequence, and of MG as a solver and MG as a preconditioner in terms of the elapsed time until the convergence. Results are presented on the NEC Cenju–3 [2].

References

[1] J.E. Dendy Jr., Black box multigrid. J. Comp. Phys., 48, 366–386 (1982).

∗C&C Research Laboratories, NEC Europe Ltd. Sankt Augustin, Germany (email: washio @ ccrl-nece.technopark.gmd.de

‡GMD, Institute for Algorithms and Scientific Computing, D-53754 Sankt Augustin, Germany (email: oosterlee @ gmd.de)

[2] R. Hempel, R. Calkin, R. Hess, W. Joppich, U. Keller, N. Koike, C.W. Oosterlee H. Ritzdorf, T. Washio, P. Wypior and W. Ziegler, Real applications on the new parallel system NEC Cenju–3. GMD Arbeitspapier 920, GMD St. Augustin, Germany (1995). To appear in Parallel Computing.

[3] T. Washio and C.W. Oosterlee, Experiences with robust parallel multilevel preconditioners for BiCGSTAB. GMD Arbeitspapier 949, GMD St. Augustin, Germany (1995). Submitted to Appl. Num. Math.

[4] P.M. de Zeeuw, Matrix-dependent prolongations and restrictions in a blackbox multigrid solver. Journ. Comp. Appl. Math., 33, 1–27 (1990).