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.
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Golub, G.H., Greif, C. & Varah, J.M. Block Orderings for Tensor-Product Grids in Two and Three Dimensions. Numerical Algorithms 30, 93–111 (2002). https://doi.org/10.1023/A:1016030016985
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DOI: https://doi.org/10.1023/A:1016030016985