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

## Authors

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

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## Abstract

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.