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Factorized sparse approximate inverse preconditionings. I: Theory. (English) Zbl 0767.65037

The present paper is the first in a series of three papers devoted to the construction of factorized, sparse, approximate inverse preconditioners and their applications to finite element schemes in the \(3D\) elasticity. In the first part, the authors present the construction techniques and investigate the properties of the preconditioners.
In particular, the preconditioners proposed preserve symmetry and/or positive definiteness of the matrix they are derived from. In case of \(M\)-, \(H\)-, or block \(H\)-matrices, the factorization method leads to convergent splittings. The authors state that this preconditioning technique is well suited for the implementation on massively parallel computers.

MSC:

65F35 Numerical computation of matrix norms, conditioning, scaling
65F10 Iterative numerical methods for linear systems
65F50 Computational methods for sparse matrices
65Y05 Parallel numerical computation
74S05 Finite element methods applied to problems in solid mechanics
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35J25 Boundary value problems for second-order elliptic equations
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