AN EVALUATION OF PARALLEL MULTIGRID AS A SOLVER AND A ...

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88 C. W. OOSTERLEE AND T. WASHIOwith jumping coefficients, can be solved efficiently. The first algorithm includes wellknownmatrix-dependent operators from the literature ([3], [6]). In [3] the operatorshave been designed so that problems on grids with arbitrary mesh sizes, not just powersof two (+1), can be solved with similar efficiency. Although in [3] these operatorsare mainly used for symmetric interface problems, we will also consider them here forunsymmetric problems.The second algorithm uses the prolongation operators introduced by de Zeeuw([25]), which are specially designed for solving unsymmetric problems. Both algorithmsemploy Galerkin coarsening ([6], [24]) for building the matrices on coarsergrids. A robust smoother is a necessary requirement in standard multigrid for achievinggrid–size-independent convergence for many problems. The alternating zebra lineGauss–Seidel relaxation method was found to be a robust smoother for 2D modelequations ([15], [24]). This smoother is used since it is possible to parallelize a linesolver efficiently on a parallel machine ([23]). Constructing robust parallel solvers isan important purpose of our work.The third multigrid algorithm in our comparison is based on the block Gaussianelimination of a matrix and an approximate Schur complement. This method has beenrecently introduced by Reusken ([12]). An interesting aspect is that the M-matrix([20]) properties of a fine grid matrix are preserved on coarse grid levels. The threemultigrid methods are described in section 2.Another robust multigrid variant for solving scalar partial differential equations isalgebraic multigrid (AMG) by Ruge and Stüben ([13]), in which the smoother is fixedbut the transfer operators depend on the connections in a matrix. Efficient parallelizationof AMG is, however, not trivial. Other multigrid preconditioners have been usedfor problems on structured grids, for example, in [1], [16], [17], and for unstructuredgrids in [2]. In [9] an early comparison of multigrid and multigrid preconditioned CGfor symmetric model equations showed the promising robustness of the latter method.A comparison between multigrid as a solver and as a preconditioner for incompressibleNavier–Stokes equations is presented in [26].The solution methods are analyzed in order to understand the convergence behaviorof the MG methods used as solvers and as preconditioners. The eigenvaluespectrum of the multigrid iteration matrices for the singularly perturbed problemsis calculated in section 3. Interesting subjects for the convergence behavior are thespectral radius and the eigenvalue distribution. Another paper in which eigenvalues ofmultigrid preconditioned Krylov methods are determined for convergence analysis is[16]. Furthermore, the polynomials that give a representation of the GMRES residualin terms of the preconditioned matrix and the initial residual are explicitly computedand solved. The evolution of the subspace in which GMRES minimizes the residualcan be determined from the solution of these polynomials.Numerical results are also presented in section 3. The benefits of the constructedmultigrid preconditioned Krylov methods are shown for 2D Poisson and convectiondiffusionproblems on fine grids solved on the NEC Cenju–3 MIMD machine ([8]).The message passing is done with the message-passing interface (MPI). Furthermore,the methods are compared with MILU ([18]) preconditioned Krylov methods.In [23] we compared several standard and nonstandard multilevel preconditionersfor BiCGSTAB. The changes in the multigrid algorithm presented in this papercan also be applied to the nonstandard multilevel approach MG-S from [23]. Furtherresearch on parallel nonstandard multilevel preconditioners particularly for 3D problems(in order to avoid alternating plane smoothers) will be described in part II ofthis paper.
MULTIGRID AS A SOLVER AND A PRECONDITIONER 8978945 612 3FIG. 1. The nine-point stencil with numbering.2. The multigrid preconditioned Krylov methods. We concentrate on linearmatrix systems with nine diagonals(1)or(2)Aφ = ba 1 i,jφ i−1,j−1 + a 2 i,jφ i,j−1 + a 3 i,jφ i+1,j−1 + a 4 i,jφ i−1,j + a 5 i,jφ i,j + a 6 i,jφ i+1,j+ a 7 i,jφ i−1,j+1 + a 8 i,jφ i,j+1 + a 9 i,jφ i+1,j+1 = b i,j ∀(i, j) ∈ G.Here, G is a subset of P nx,ny = {(i, j)|1 ≤ i ≤ n x , 1 ≤ j ≤ n y }.The stencil is presented in Figure 1 for convenience.Matrix A has right preconditioning as follows:(3)AK −1 (Kφ) = b.The Krylov subspace methods that are used for solving (3) are BiCGSTAB ([19]), asdescribed in detail in [23], and GMRES(m) ([14]).2.1. GMRES. The GMRES(m) algorithm with a right preconditioner appearsas follows:GMRES (m,A, b, φ) {Choose φ (0) , dimension m, matrix ˜H = 0 with dim: (m + 1) × mr (0) = b − Aφ (0) ; β = ||r (0) || 2 ; f 1 = r (0) /β;for j = 1, . . . , m {u j := K −1 f j ;w := Au j ;for i = 1, . . . , j {h i,j := (w, f i );w := w − h i,j f j ;h j+1,j := ||w|| 2 , f j+1 = w/h j+1,j ;}}Define F m := [f 1 , . . . , f m ];φ (m) := φ (0) + K −1 F m y m ; with y m = min y ||βe 1 − ˜Hy|| 2 ,(e 1 = [1, 0, . . . , 0] T );Compute r (m) = b − Aφ (m) ;If satisfied stop, else restart φ (0) ← φ (m) ;}K −1 f j is the preconditioning step, which is one iteration of a multigrid cycle.
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AN EVALUATION OF PARALLEL MULTIGRID AS A SOLVER AND A ...

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