Block Orderings for Tensor-Product Grids in Two and Three Dimensions
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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.
- S. Barnett, Matrices-Methods and Applications (Clarendon Press, Oxford, 1990).
- H.C. Elman and G.H. Golub, Iterative methods for cyclically reduced non-self-adjoint linear systems, Math. Comp. 54 (1990) 671–700.
- H.C. Elman and G.H. Golub, Iterative methods for cyclically reduced non-self-adjoint linear systems II, Math. Comp. 56 (1991) 215–242.
- G.H. Golub and D. Vanderstraeten, On the preconditioning of matrices with skew-symmetric splittings, Numer. Algorithms 25 (2000) 223–239.
- G.H. Golub and C.F. Van Loan, Matrix Computations, 3rd ed. (Johns Hopkins Univ. Press, Baltimore, MD, 1996).
- C. Greif, Cyclic reduction for three-dimensional elliptic equations with variable coefficients, SIAM J. Matrix Anal. Appl. 21(1) (1999) 29–44.
- C. Greif and J.M. Varah, Iterative solution of cyclically reduced systems arising from discretization of the three-dimensional convection-diffusion equation, SIAMJ. Sci. Comput. 19(6) (1998) 1018–1040.
- C. Greif and J.M. Varah, Block stationary methods for nonsymmetric cyclically reduced systems arising from three-dimensional elliptic equations, SIAM J. Matrix Anal. Appl. 20(4) (1999) 1038–1059.
- R.A. Horn and C.R. Johnson, Matrix Analysis (Cambridge Univ. Press, Cambridge, 1985).
- H. Melbø, Preconditioning and error estimates for iterative linear solvers, Ph.D. thesis, Studieretning for industriell Matematikk, NTNU (1998).
- S.V. Parter, On 'two-line' iterative methods for the Laplace and biharmonic difference equations, Numer. Math. 11 (1959) 240–252.
- Y. Saad, Iterative Methods for Sparse Linear Systems (PWS Publishing, Boston, 1996).
- R.S. Varga, Matrix Iterative Analysis (Prentice-Hall, Englewood Cliffs, NJ, 1962).
- H.A. Van Der Vorst, Bi-CGSTAB: A fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric linear systems, SIAM J. Sci. Statist. Comput. 13 (1992) 631–644.
- D.M. Young, Iterative Solution of Large Linear Systems (Academic Press, New York, 1971).
- Block Orderings for Tensor-Product Grids in Two and Three Dimensions
Volume 30, Issue 2 , pp 93-111
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- Kluwer Academic Publishers
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- sparse linear systems
- discretization of PDEs
- convergence of iterative solvers
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- Author Affiliations
- 1. SCCM Program, Gates 2B, Stanford University, Stanford, CA, 94305, USA
- 2. Parametric Technology Corporation, 2590 North First St., Suite 200, San Jose, CA, 95131, USA
- 3. Department of Computer Science, University of British Columbia, Vancouver, BC, V6T 1Z4, Canada