Summary. We derive new estimates for the rate of convergence of the conjugate gradient method by utilizing isolated eigenvalues of parts of the spectrum. We present a new generalized version of an incomplete factorization method and compare the derived estimates of the number of iterations with the number actually found for some elliptic difference equations and for a similar problem with a model empirical distribution function.

Summary. A class of preconditioning methods depending on a relaxation parameter is presented for the solution of large linear systems of equations Ax=b, where A is a symmetric positive definite matrix. The methods are based on an incomplete factorization of the matrix A and include both pointwise and blockwise factorizations. We study the dependence of the rate of convergence of the preconditioned conjugate gradient method on the distribution of eigenvalues of C-1A, where C is the preconditioning matrix. We also show graphic representations of the eigenvalues and present numerical tests of the methods.

The discretized linear elasticity problem is solved by the preconditioned conjugate gradient (pcg) method. Mainly we consider the linear isotropic case but we also comment on the more general linear orthotropic problem. The preconditioner is based on the separate displacement component (sdc) part of the equations of elasticity. The preconditioning system consists of two or three subsystems (in two or three dimensions) also called inner systems, each of which is solved by the incomplete factorization pcg‐method, i.e., we perform inner iterations. A finite element discretization and node numbering giving a high degree of partial parallelism with equal processor load for the solution of these systems by the MIC(0) pcg method is presented. In general, the incomplete factorization requires an M‐matrix. This property is studied for the elasticity problem. The rate of convergence of the pcg‐method is analysed for different preconditionings based on the sdc‐part of the elasticity equations. In the following two parts of this trilogy we will focus more on parallelism and implementation aspects. © 1998 John Wiley & Sons, Ltd.

SUMMARYThis is the second part of a trilogy on parallel solution of the linear elasticity problem. We consider the plain case of the problem with isotropic material, including discontinuous coe cients, and with homogeneous Dirichlet boundary condition. The discretized problem is solved by the preconditioned conjugate gradient (pcg) method.In the ÿrst part of the trilogy block-diagonal preconditioners based on the separate displacement component part of the elasticity equations were analysed. The preconditioning systems were solved by the pcg-method, i.e. inner iterations were performed. As preconditioner, we used modiÿed incomplete factorization MIC(0), where possibly the element matrices were modiÿed in order to give M -matrices, i.e. in order to guarantee the existence of the MIC(0) factorization.In the present paper, the second part, full block incomplete factorization preconditioners are presented and analysed. In order to avoid inner=outer iterations we also study a variant of the block-diagonal method and of the full block method, where the matrices of the inner systems are just replaced by their MIC(0)-factors. A comparison is made between the various methods with respect to rate of convergence and work per unknown. The fastest methods are implemented by message passing utilizing the MPI system.In the third part of the trilogy, we will focus on the use of higher-order ÿnite elements.

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