Maya Neytcheva

Eigenvalue estimates for preconditioned saddle point matrices

Eigenvalue bounds for saddle point matrices on symmetric or, more generally, nonsymmetric form are derived and applied for preconditioned versions of the matrices. The preconditioners enable efficient iterative solution of the corresponding linear systems.

Preconditioning methods for linear systems arising in constrained optimization problems

Preconditioning methods for matrices on saddle point form, as typically arising in equality constrained optimization problems, are surveyed. Special consideration is given to two methods: a nearly symmetric block incomplete factorization preconditioning method and a preconditioner on the same saddle point form as the given matrix. Both methods result in eigenvalues with positive real parts and small or zero imaginary parts. The behaviour of the methods are illustrated by solving a regularized Stokes problem. Copyright

A comparison of iterative methods to solve complex valued linear algebraic systems

Complex valued linear algebraic systems arise in many important applications. We present analytical and extensive numerical comparisons of some available numerical solution methods. It is advocated, in particular for large scale ill-conditioned problems, to rewrite the complex-valued system in real valued form leading to a two-by-two block system of particular form, for which it is shown that a very efficient and robust preconditioned iterative solution method can be constructed. Alternatively, in many cases it turns out that a simple preconditioner in the form of the sum of the real and the imaginary part of the matrix also works well but involves complex arithmetic.

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 (

Efficient preconditioners for large scale binary Cahn–Hilliard models

m

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

) than the customary

Solving the Stokes Problem on a Massively Parallel Computer

m=2

Numerical and computational efficiency of solvers for two-phase problems

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

Robust optimal multilevel preconditioners for non‐conforming finite element systems

Abstract In this work we consider preconditioned iterative solution methods for numerical simulations of multiphase flow problems, modelled by the Cahn-Hilliard equation. We focus on diphasic flows and the construction and efficiency of a preconditioner for the algebraic systems arising from finite element discretizations in space and the θ-method in time. The preconditioner utilises to a full extent the algebraic structure of the underlying matrices and exhibits optimal convergence and computational complexity properties. Various numerical experiments, including large scale examples, are presented as well as performance comparisons with other solution methods.

A general approach to analyse preconditioners for two‐by‐two block matrices

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