Robert Beauwens

HIGH-PERFORMANCE PCG SOLVERS FOR FEM STRUCTURAL ANALYSIS

The preconditioned conjugate gradient algorithm is a well-known and powerful method used to solve large sparse symmetric positive definite linear systems Such system are generated by the finite element discretisation in structural analysis but users of finite elements in this contest generally still rely on direct methods It is our purpose in the present work to highlight the improvement brought forward by some new preconditioning techniques and show that the preconditioned conjugate gradient method performs better than efficient direct methods.

Upper eigenvalue bounds for pencils of matrices

Abstract Eigenvalue bounds are obtained for pencils of matrices A − vB where A is a Stieltjes matrix and B is positive definite, under assumptions suitable for the estimation of asymptotic convergence rates of factorization iterative methods, where B represents the approximate factorization of A . The upper bounds obtained depend on the “connectivity” structure of the matrices involved, which enters through matrix graph considerations; in addition, a more classical argument is used to obtain a lower bound. Potential applications of these results include a partial confirmation of Gustafssons conjecture concerning the nonnecessity of Axelssons perturbations.

Preconditioning of discrete Helmholtz operators perturbed by a diagonal complex matrix

Incomplete factorizations are popular preconditioning techniques for solving large and sparse linear systems. In the case of highly indefinite complex–symmetric linear systems, the convergence of Krylov subspace methods sometimes degrades with increasing level of fill-in. The reasons for this disappointing behaviour are twofold. On the one hand, the eigenvalues of the preconditioned system tend to 1, but the ‘convergence’ is not monotonous. On the other hand, the eigenvalues with negative real part, on their move towards 1 have to cross the origin, whence the risk of clustering eigenvalues around 0 while ‘improving’ the preconditioner. This makes it risky to predict any gain when passing from a level to a higher one. We examine a remedy which consists in slightly moving the spectrum of the original system matrix along the imaginary axis. Theoretical analysis that motivates our approach and experimental results are presented, which displays the efficiency of the new preconditioning techniques. Copyright

On Axelsson's perturbations

Abstract A nice perturbation technique was introduced by Axelsson and further developed by Gustafsson to prove that factorization iterative methods are able, under appropriate conditions, to reach a convergence rate larger by an order of magnitude than that of classical schemes. Gustafsson observed however that the perturbations introduced to prove this result seemed actually unnecessary to reach it in practice. In the present work, on the basis of eigenvalue bounds recently obtained by the author, we offer an alternative approach which brings a partial confirmation of Gustafssons conjecture.

Conditioning analysis of positive definite matrices by approximate factorizations

Abstract The conditioning analysis of positive definite matrices by approximate LU factorizations is usually reduced to that of Stieltjes matrices (or even to more specific classes of matrices) by means of perturbation arguments like spectral equivalence. We show in the present work that a wider class, which we call “almost Stieltjes” matrices, can be used as reference class and that it has decisive advantages for the conditioning analysis of finite element approximations of large multidimensional steady-state diffusion problems.

On Sparse Block Factorization Iterative Methods

A synthetic formalism is proposed for the description of sparse block factorization iterative methods. It is used to develop existence, convergence and comparison theorems and to compare block factorization schemes against point factorization schemes.

Factorization iterative methods,M-operators andH-operators

SummaryWe set up here a general formalism for describing factorization iterative methods of the first order and we use it to review various methods that have been proposed in the literature; next we introduce the notions ofM- andH-operators which generalize those of block-M- and block-H-matrices; finally we discuss the properties of factorization iterative methods in relation with characteristic properties ofM- andH-operators.

Semistrict diagonal dominance

The notion of semistrict diagonal dominance is introduced and shown to provide a means for separating the properties of diagonally dominant matrices which depend on irreducibility from those which do not. Refinements of well-known monotonicity criteria are obtained as applications.

EXISTENCE AND CONDITIONING PROPERTIES OF SPARSE APPROXIMATE BLOCK FACTORIZATIONS

A particular class of sparse approximate block

Problem-dependent preconditioners for iterative solvers in FE elastostatics

LU