Radim Blaheta

Displacement decomposition—incomplete factorization preconditioning techniques for linear elasticity problems

Two preconditioning techniques for the numerical solution of linear elasticity problems are described and studied. Both techniques are based on spectral equivalence approach. The first technique consists in an incomplete factorization of the separate displacement component part of the stiffness matrix. The second technique uses an incomplete factorization of the isotropic approximation to the stiffness matrix. Results concerning existence, stability and efficiency of these preconditioning techniques are presented. The efficiency and robustness of the described techniques are illustrated by numerical experiments.

Preconditioning of Boundary Value Problems Using Elementwise Schur Complements

This paper deals with an efficient technique for computing high-quality approximations of Schur complement matrices to be used in various preconditioners for the iterative solution of finite element discretizations of elliptic boundary value problems. The Schur complements are based on a two-by-two block decomposition of the matrix, and their approximations are computed by assembly of local (macroelement) Schur complements. The block partitioning is done by imposing a particular node ordering following the grid refinement hierarchy in the discretization mesh. For the theoretical derivation of condition number bounds, but not for the actual application of the method, we assume that the corresponding differential operator is self-adjoint and positive definite. The numerical efficiency of the proposed Schur complement approximation is illustrated in the framework of block incomplete factorization preconditioners of multilevel type, which require approximations of a sequence of arising Schur complement matrices. The behavior of the proposed approximation is compared with that of the coarse mesh finite element matrix, commonly used as an approximation of the Schur complement in the context of the above preconditioning methods. Moreover, the influence of refining a coarse mesh using a higher refinement number (

Convergence of Newton-type methods in incremental return mapping analysis of elasto-plastic problems

m

GPCG–generalized preconditioned CG method and its use with non‐linear and non‐symmetric displacement decomposition preconditioners

) than the customary

Uniform estimate of the constant in the strengthened CBS inequality for anisotropic non‐conforming FEM systems

m=2

Robust optimal multilevel preconditioners for non‐conforming finite element systems

is analyzed and its efficiency is also illustrated by numerical tests.

Preconditioning of matrices partitioned in 2 × 2 block form: eigenvalue estimates and Schwarz DD for mixed FEM

Abstract The incremental finite element algorithm with return mapping stress computation is considered for the solution of problems of elasto-plasticity. This algorithm, performed in load steps, leads to the necessity of solving large scale nonlinear systems. The properties of these systems are investigated in the presented paper together with two iterative techniques for their solution. The main results of the paper are rigorous proofs of the convergence of the inexact initial stiffness method and the local quadratic convergence of the inexact Newton method with the consistent tangent operator.

Stable discretization of poroelasticity problems and efficient preconditioners for arising saddle point type matrices

The paper investigates a generalization of the preconditioned conjugate gradient method, which uses explicit orthogonalization of the search directions. This generalized preconditioned conjugate gradient (GPCG) method is suitable for solving the symmetric positive definite systems with preconditioners, which can be non-symmetric or non-linear. Such preconditioners can arise from computing of the pseudoresiduals by some additive or multiplicative space decomposition method with inexact solution of the auxiliary subproblems by inner iterations. The non-linear and non-symmetric preconditioners based on displacement decomposition for solving elasticity problems are described as an example of such preconditioners. Copyright

Schwarz methods for discrete elliptic and parabolic problems with an application to nuclear waste repository modelling

Uniform estimate of the constant in the strengthened CBS inequality for anisotropic non-conforming FEM systems

Large scale parallel FEM computations of far/near stress field changes in rocks

We consider strategies to construct optimal order two- and multilevel hierarchical preconditioners for linear systems as arising from the finite element discretization of self-adjoint second order elliptic problems using non-conforming Crouzeix–Raviart linear elements. In this paper we utilize the hierarchical decompositions, derived in a previous work by the same authors (Numerical Linear Algebra with Applications 2004; 11:309–326) and provide a further analysis of these decompositions in order to assure robustness with respect to anisotropy. Finally, we show how to construct both multiplicative and additive versions of the algebraic multilevel iteration preconditioners and show robustness and optimal order convergence estimates. Copyright