The Deflated Preconditioned Conjugate Gradient Method Applied to Composite Materials

Simulations with composite materials often involve large jumps in the coefficients of the underlying stiffness matrix. These jumps can introduce unfavorable eigenvalues in the spectrum of the stiffness matrix. We show that the rigid body modes; the translations and rotations, of the disjunct rigid bodies in the composite material correspond to...

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...

Efficient Multigrid Methods based on Improved Coarse Grid Correction Techniques

Multigrid efficiency often suffers from inadequate coarse grid correction in different prototypic situations. We select a few problems, where coarse grid correction issues arise because of anisotropic coefficients, non-equidistant or non-uniform grid stretching, or inherent indefiniteness in the partial differential equation. Most of the work in...

Two-level preconditioned conjugate gradient methods with applications to bubbly flow problems

The Preconditioned Conjugate Gradient (PCG) method is one of the most popular iterative methods for solving large linear systems with a symmetric and positive semi-definite coefficient matrix. However, if the preconditioned coefficient matrix is ill-conditioned, the convergence of the PCG method typically deteriorates. Instead, a two-level PCG...

On a decoupled algorithm for poroelasticity and its resolution by multigrid

In this work, we present a robust and efficient multigrid solver for a reformulated version of the system of poroelasticity equations. The reformulation enables us to treat the system in a decoupled fashion. It is sufficient to choose a highly efficient multigrid method for a scalar Poisson type equation for the overall solution of this...

Multigrid for high dimensional elliptic partial differential equations on non-equidistant grids

This work presents techniques, theory and numbers for multigrid in a general d-dimensional setting. The main focus is the multigrid convergence for high-dimensional partial differential equations (PDEs). As a model problem we have chosen the anisotropic diffusion equation, on a unit hypercube. We present some techniques for building the general...

Some numerical aspects for solving sparse large linear systems derived from the Helmholtz equation

In this report, several numerical aspects and diculties for solving a linear system derived from the time-harmonic wave equations are overviewed. The presentation begins with the derivation of the governing equation for waves propagating in general inhomogeneous media. Due to the need of numerical solutions, various discretizations based on...

Multi signal-domain modeling of solid-state transducers: A theoretical framework based on irreversible thermodynamics mixed finite elements and multigrid