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Iterative Methods for Cyclically Reduced Non-Self-Adjoint Linear Systems

Howard C. Elman and Gene H. Golub
Mathematics of Computation
Vol. 54, No. 190 (Apr., 1990), pp. 671-700
DOI: 10.2307/2008506
Stable URL: http://www.jstor.org/stable/2008506
Page Count: 30
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Iterative Methods for Cyclically Reduced Non-Self-Adjoint Linear Systems
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Abstract

We study iterative methods for solving linear systems of the type arising from two-cyclic discretizations of non-self-adjoint two-dimensional elliptic partial differential equations. A prototype is the convection-diffusion equation. The methods consist of applying one step of cyclic reduction, resulting in a "reduced system" of half the order of the original discrete problem, combined with a reordering and a block iterative technique for solving the reduced system. For constant-coefficient problems, we present analytic bounds on the spectral radii of the iteration matrices in terms of cell Reynolds numbers that show the methods to be rapidly convergent. In addition, we describe numerical experiments that supplement the analysis and that indicate that the methods compare favorably with methods for solving the "unreduced" system.

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