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The efficient solution to nonsymmetric linear systems is still an open issue, especially on parallel computers. In this paper we generalize to the unsymmetric case the Block Factorized Sparse Approximate Inverse (Block FSAI) preconditioner which has already proved very effective on symmetric positive definite (SPD) problems. Block FSAI is a hybrid approach combining an ‘‘inner’’ preconditioner, with the aim of transforming the system matrix structure to block diagonal, with an ‘‘outer’’ one, a block diagonal incomplete or exact factorization intended to improve the conditioning of each block. The proposed algorithm is experimented with in a number of large size matrices showing both a good robustness and scalability

SUMMARYPreconditioned projection (or conjugate gradient like) methods are increasingly used for the accurate and efficient solution to finite element (FE) coupled consolidation equations. Theory indicates that preliminary row/column scaling does not affect the eigenspectrum of the iteration matrix controlling convergence as long as the preconditioner relies on the incomplete factorization of the FE coefficient matrix. However, computational experience with mid-large size problems shows that the above inexpensive operation can significantly accelerate the solver convergence, and to a minor extent also improve the final accuracy, as a result of a better solver stability to the accumulation and propagation of floating point round-off errors. This is demonstrated with the aid of the least square logarithm (LSL) scaling algorithm on FE consolidation problems of increasing size up to more than 100 000. It is shown that a major source of numerical instability rests with the sub-matrix which couples the structural to the fluid part of the underlying mathematical model. It is concluded that for mid-large size, possibly difficult, FE consolidation problems left/right LSL scaling is to be always recommended when the incomplete factorization is used as a preconditioning technique.

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The meshless local Petrov-Galerkin (MLPG) method is a meshfree procedure for solving partial differential equations. However, the benefit in avoiding the mesh construction and refinement is counterbalanced by the use of complicated non polynomial shape functions with subsequent difficulties, and a potentially large cost, when implementing numerical integration schemes. In this paper we describe and compare some numerical quadrature rules with the aim at preserving the MLPG solution accuracy and at the same time reducing its computational cost.