J.M. Tang

Comparison of Two-Level Preconditioners Derived from Deflation, Domain Decomposition and Multigrid Methods

For various applications, it is well-known that a multi-level, in particular two-level, preconditioned CG (PCG) method is an efficient method for solving large and sparse linear systems with a coefficient matrix that is symmetric positive definite. The corresponding two-level preconditioner combines traditional and projection-type preconditioners to get rid of the effect of both small and large eigenvalues of the coefficient matrix. In the literature, various two-level PCG methods are known, coming from the fields of deflation, domain decomposition and multigrid. Even though these two-level methods differ a lot in their specific components, it can be shown that from an abstract point of view they are closely related to each other. We investigate their equivalences, robustness, spectral and convergence properties, by accounting for their implementation, the effect of roundoff errors and their sensitivity to inexact coarse solves, severe termination criteria and perturbed starting vectors.

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 projection-type preconditioner to get rid of the effect of both small and large eigenvalues of the coefficient matrix; multilevel approaches arise by recursively applying the two-level technique within the projection step. In the literature, various such preconditioners are known, coming from the fields of deflation, domain decomposition, and multigrid (MG). At first glance, these methods seem to be quite distinct; however, from an abstract point of view, they are closely related. The aim of this paper is to relate two-level PCG methods with symmetric two-grid (V(1,1)-cycle) preconditioners (derived from MG approaches), in their abstract form, to deflation methods and a two-level domain-decomposition approach inspired by the balancing Neumann-Neumann method. The MG-based preconditioner is often expected to be more effective than these other two-level preconditioners, but this is shown to be not always true. For common choices of the parameters, MG leads to larger error reductions in each iteration, but the work per iteration is more expensive, which makes this comparison unfair. We show that, for special choices of the underlying one-level preconditioners in the deflation or domain-decomposition methods, the work per iteration of these preconditioners is approximately the same as that for the MG preconditioner, and the convergence properties of the resulting two-level PCG methods will also be (approximately) the same. This means that, in this respect, the particular choice of the two-level preconditioner is less important than the choice of the parameters. Numerical experiments are presented to emphasize the theoretical results.

Fast and robust solvers for pressure-correction in bubbly flow problems

We consider the numerical simulation of two-phase fluid flow, where bubbles or droplets of one phase move against a background of the other phase. Such flows are governed by the Navier-Stokes equations, the solution of which may be approximated using a pressure-correction approach. Within such an approach, the computational cost is often dominated by the solution of a linear system corresponding to a discrete Poisson equation with discontinuous coefficients. In this paper, we explore the efficient solution of these linear systems using robust multilevel solvers, such as deflated variants of the preconditioned conjugate gradient method, or robust multigrid techniques. We consider these families of methods in more detail and compare their performance in the simulation of bubbly flows. Some of these methods turn out to be very effective and reduce the amount of work to solve the pressure-correction system substantially, resulting in efficient calculations for two-phase flows on highly resolved grids.

Acceleration of preconditioned Krylov solvers for bubbly flow problems

We consider the linear system arising from discretization of the pressure Poisson equation with Neumann boundary conditions, derived from bubbly flow problems. In the literature, preconditioned Krylov iterative solvers are proposed, but they often suffer from slow convergence for relatively large and complex problems. We extend these traditional solvers with the so-called deflation technique, that accelerates the convergence substantially and has favorable parallel properties. Several numerical aspects are considered, such as the singularity of the coefficient matrix and the varying density field at each time step. We demonstrate theoretically that the resulting deflation method accelerates the convergence of the iterative process. Thereafter, this is also demonstrated numerically for 3-D bubbly flow applications, both with respect to the number of iterations and the computing time.

Acceleration of Preconditioned Krylov Solvers for Bubbly Flow Problems

We consider the linear system arising from discretization of the pressure Poisson equation with Neumann boundary conditions, derived from bubbly flow problems. In the literature, preconditioned Krylov iterative solvers are proposed, but they often suffer from slow convergence for relatively large and complex problems. We extend these traditional solvers with the so-called deflation technique, that accelerates the convergence substantially and has favorable parallel properties. Several numerical aspects are considered, such as the singularity of the coefficient matrix and the varying density field at each time step. We demonstrate theoretically that the resulting deflation method accelerates the convergence of the iterative process. Thereafter, this is also demonstrated numerically for 3-D bubbly flow applications, both with respect to the number of iterations and the computing time.