Numerical Algorithms

, Volume 30, Issue 2, pp 93–111

Block Orderings for Tensor-Product Grids in Two and Three Dimensions


  • Gene H. Golub
    • SCCM Program, Gates 2BStanford University
  • Chen Greif
    • Parametric Technology Corporation
  • James M. Varah
    • Department of Computer ScienceUniversity of British Columbia

DOI: 10.1023/A:1016030016985

Cite this article as:
Golub, G.H., Greif, C. & Varah, J.M. Numerical Algorithms (2002) 30: 93. doi:10.1023/A:1016030016985


We consider two-line and two-plane orderings for a convection–diffusion model problem in two and three dimensions, respectively. These strategies are aimed at introducing dense diagonal blocks, at the price of a slight increase of the bandwidth of the matrix, compared to natural lexicographic ordering. Comprehensive convergence analysis is performed for the block Jacobi scheme. We then move to consider a two-step preconditioning technique, and analyze the numerical properties of the linear systems that are solved in each step of the iterative process. For the 3-dimensional problem this approach is a viable alternative to the Incomplete LU approach, and may be easier to implement in parallel environments. The analysis is illustrated and validated by numerical examples.

sparse linear systemsdiscretization of PDEsorderingsconvergence of iterative solvers

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© Kluwer Academic Publishers 2002