On the use of rigid body modes in the deflated preconditioned conjugate gradient method

Large discontinuities in material properties, such as those encountered in composite materials, lead to ill-conditioned systems of linear equations. These discontinuities give rise to small eigenvalues that may negatively affect the convergence of iterative solution methods such as the preconditioned conjugate gradient method. This paper...

GPU implementation of deflated preconditioned conjugate gradient

A Comparison of Two-Level Preconditioners Based on Multigrid and Deflation

It is well known that two-level and multilevel preconditioned conjugate gradient (PCG) methods provide efficient techniques for solving large and sparse linear systems whose coefficient matrices are symmetric and positive definite. A two-level PCG method combines a traditional (one-level) preconditioner, such as incomplete Cholesky, with a...

Domain decomposition methods and deflated Krylov subspace iterations

The balancing Neumann-Neumann (BNN) and the additive coarse grid correction (BPS) preconditioner are fast and successful preconditioners within domain decomposition methods for solving partial differential equations. For certain elliptic problems these preconditioners lead to condition numbers which are independent of the mesh sizes and are...

Deflated ICCG Method solving the Singular and Discontinuous Diffusion Equation derived from 3-D Multi-Phase Flows

Simulating bubbly flows is a very popular topic in CFD. These bubbly flows are governed by the Navier-Stokes equations. In many popular operator splitting formulations for these equations, solving the linear system coming from the discontinuous diffusion equation takes the most computational time, despite of its elliptic origins. Sometimes these...

New variants of deflation techniques for bubbly flow problems

For various applications, it is well-known that deflated ICCG is an efficient method for solving linear systems with invertible and singular co-efficient matrix. This deflated ICCG with subdomain deflation vectors is used by us to solve linear systems with singular coefficient matrix, arising from a discretization of the Poisson equation with...

On the theory of deflation and singular symmetric positive semi-definite matrices

In this report we give new insights into the properties of invertible and singular deflated and preconditioned linear systems where the coefficient matrices are also symmetric and positive (semi-) definite. First we prove that the invertible de ated matrix has always a more favorable effeective condition number compared to the original matrix....

A comparison of deflation and the balancing Neumann-Neumann preconditioner

In this paper we compare various preconditioners for the numerical solution of partial differential equations. We compare the well-known balancing Neumann Neumann preconditioner used in domain decomposition methods with a so-called deflation preconditioner. We prove that the effective condition number of the deflated preconditioned system is...

A comparison of deflation and coarse grid correction applied to porous media flow

In this paper we compare various preconditioners for the numerical solution of partial dierential equations. We compare a coarse grid correction preconditioner used in domain decomposition methods with a so-called deflation preconditioner. We prove that the effective condition number of the de ated preconditioned system is always, i.e. for all...

Deflation in preconditioned conjugate gradient methods for Finite Element Problems

We investigate the influence of the value of deflation vectors at interfaces on the rate of convergence of preconditioned conjugate gradient methods applied to a Finite Element discretization for an elliptic equation. Our set-up is a Poisson problem in two dimensions with continuous or discontinuous coefficients that vary in several orders of...

A comparison of various deflation vectors applied to elliptic problems with discontinuous coefficients

The influence of deflation vectors at interfaces on the deflated conjugate gradient method

We investigate the influence of the value of deflation vectors at interfaces on the rate of convergence of preconditioned conjugate gradient methods. Our set-up is a Laplace problem in two dimensions with continuous or discontinuous coeffcients that vary in several orders of magnitude. In the continuous case we are interested in the convergence...