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Approximate and incomplete factorizations. (English) Zbl 0865.65015
Keyes, David E. (ed.) et al., Parallel numerical algorithms. Proceedings of the workshop, Hampton, VA, May 23–25, 1994. Dordrecht: Kluwer Academic Publishers. ICASE/LaRC Interdisciplinary Series in Science and Engineering. 4, 167-202 (1997).
We give a brief overview of a particular class of preconditioners known as incomplete factorizations. They can be thought of as approximating the exact LU factorization of a given matrix \(A\) (e.g., computed via Gaussian elimination) by disallowing certain fill-ins. As opposed to other partial differential equation (PDE)-based preconditioners such as multigrid and domain decomposition, this class of preconditioners is primarily algebraic in nature and can in principle be applied to general sparse matrices. When applied to PDE problems, they are usually not optimal in the sense that the condition number of the preconditioned system grows as the mesh size \(h\) is reduced, although usually at a slower rate than for the unpreconditioned system. On the other hand, they are often quite robust with respect to other more algebraic features of the problem such as rough and anisotropic coefficients and strong convection terms.
For the entire collection see [Zbl 0857.00034].

65F10 Iterative numerical methods for linear systems
65F50 Computational methods for sparse matrices
65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs
65F35 Numerical computation of matrix norms, conditioning, scaling