Results 1  10
of
56
Numerical solution of saddle point problems
 ACTA NUMERICA
, 2005
"... Large linear systems of saddle point type arise in a wide variety of applications throughout computational science and engineering. Due to their indefiniteness and often poor spectral properties, such linear systems represent a significant challenge for solver developers. In recent years there has b ..."
Abstract

Cited by 180 (30 self)
 Add to MetaCart
Large linear systems of saddle point type arise in a wide variety of applications throughout computational science and engineering. Due to their indefiniteness and often poor spectral properties, such linear systems represent a significant challenge for solver developers. In recent years there has been a surge of interest in saddle point problems, and numerous solution techniques have been proposed for solving this type of systems. The aim of this paper is to present and discuss a large selection of solution methods for linear systems in saddle point form, with an emphasis on iterative methods for large and sparse problems.
Performance and analysis of saddle point preconditioners for the discrete steadystate NavierStokes equations
 NUMER. MATH. (2002) 90: 665–688
, 2002
"... ..."
Efficient Preconditioning Of The Linearized NavierStokes Equations
 J. Comp. Appl. Math
, 1999
"... We outline a new class of robust and efficient methods for solving subproblems that arise in the linearization and operator splitting of NavierStokes equations. We describe a very general strategy for preconditioning that has two basic building blocks; a multigrid Vcycle for the scalar convection ..."
Abstract

Cited by 54 (14 self)
 Add to MetaCart
We outline a new class of robust and efficient methods for solving subproblems that arise in the linearization and operator splitting of NavierStokes equations. We describe a very general strategy for preconditioning that has two basic building blocks; a multigrid Vcycle for the scalar convectiondiffusion operator, and a multigrid Vcycle for a pressure Poisson operator. We present numerical experiments illustrating that a simple implementation of our approach leads to an effective and robust solver strategy in that the convergence rate is independent of the grid, robust with respect to the timestep, and only deteriorates very slowly as the Reynolds number is increased.
An augmented Lagrangianbased approach to the Oseen problem
 SIAM J. Sci. Comput
, 2006
"... Abstract. We describe an effective solver for the discrete Oseen problem based on an augmented Lagrangian formulation of the corresponding saddle point system. The proposed method is a block triangular preconditioner used with a Krylov subspace iteration like BiCGStab. The crucial ingredient is a no ..."
Abstract

Cited by 52 (23 self)
 Add to MetaCart
Abstract. We describe an effective solver for the discrete Oseen problem based on an augmented Lagrangian formulation of the corresponding saddle point system. The proposed method is a block triangular preconditioner used with a Krylov subspace iteration like BiCGStab. The crucial ingredient is a novel multigrid approach for the (1,1) block, which extends a technique introduced by Schöberl for elasticity problems to nonsymmetric problems. Our analysis indicates that this approach results in fast convergence, independent of the mesh size and largely insensitive to the viscosity. We present experimental evidence for both isoP2P0 and isoP2P1 finite elements in support of our conclusions. We also show results of a comparison with two stateoftheart preconditioners, showing the competitiveness of our approach. Key words. Navier–Stokes equations, finite element, iterative methods, multigrid, preconditioning AMS subject classifications. 65F10, 65N22, 65F50 DOI. 10.1137/050646421 1. Introduction. We consider the numerical solution of the steady Navier– Stokes equations governing the flow of a Newtonian, incompressible viscous fluid. Let Ω ⊂ R d (d =2,3) be a bounded, connected domain with a piecewise smooth
Preconditioning For The SteadyState NavierStokes Equations With Low Viscosity
 SIAM J. SCI. COMPUT
, 1996
"... We introduce a preconditioner for the linearized NavierStokes equations that is effective when either the discretization mesh size or the viscosity approaches zero. For constant coefficient problems with periodic boundary conditions, we show that the preconditioning yields a system with a single ei ..."
Abstract

Cited by 50 (10 self)
 Add to MetaCart
We introduce a preconditioner for the linearized NavierStokes equations that is effective when either the discretization mesh size or the viscosity approaches zero. For constant coefficient problems with periodic boundary conditions, we show that the preconditioning yields a system with a single eigenvalue equal to one, so that performance is independent of both viscosity and mesh size. For other boundary conditions, we demonstrate empirically that convergence depends only mildly on these parameters and we give a partial analysis of this phenomenon. We also show that some expensive subsidiary computations required by the new method can be replaced by inexpensive approximate versions of these tasks based on iteration, with virtually no degradation of performance.
Approximate Inverse Techniques for BlockPartitioned Matrices
 SIAM J. Sci. Comput
, 1995
"... This paper proposes some preconditioning options when the system matrix is in blockpartitioned form. This form may arise naturally, for example from the incompressible NavierStokes equations, or may be imposed after a domain decomposition reordering. Approximate inverse techniques are used to g ..."
Abstract

Cited by 43 (12 self)
 Add to MetaCart
This paper proposes some preconditioning options when the system matrix is in blockpartitioned form. This form may arise naturally, for example from the incompressible NavierStokes equations, or may be imposed after a domain decomposition reordering. Approximate inverse techniques are used to generate sparse approximate solutions whenever these are needed in forming the preconditioner. The storage requirements for these preconditioners may be much less than for ILU preconditioners for tough, largescale CFD problems. The numerical experiments reported show that these preconditioners can help us solve difficult linear systems whose coefficient matrices are highly indefinite. 1 Introduction Consider the block partitioning of a matrix A, in the form A = ` B F E C ' (1) where the blocking naturally occurs due the ordering of the equations and the variables. Matrices of this form arise in many applications, such as in the incompressible NavierStokes equations, where the sc...
Structured backward error and condition of generalized eigenvalue problems
 SIAM J. Matrix Anal. Appl
, 1998
"... Abstract. Backward errors and condition numbers are defined and evaluated for eigenvalues and eigenvectors of generalized eigenvalue problems. Both normwise and componentwise measures are used. Unstructured problems are considered first, and then the basic definitions are extended so that linear str ..."
Abstract

Cited by 29 (4 self)
 Add to MetaCart
Abstract. Backward errors and condition numbers are defined and evaluated for eigenvalues and eigenvectors of generalized eigenvalue problems. Both normwise and componentwise measures are used. Unstructured problems are considered first, and then the basic definitions are extended so that linear structure in the coefficient matrices (for example, Hermitian, Toeplitz, Hamiltonian, or band structure) is preserved by the perturbations.
On Solving BlockStructured Indefinite Linear Systems
 SIAM J. Sci. Comput
, 2003
"... We consider 2 × 2 block indefinite linear systems whose (2,2) block is zero. Such systems arise in many applications. We focus on techniques that change the linear systems in a way that may make it easier to solve them. In particular, two techniques based on modifying the (1,1) block are consi ..."
Abstract

Cited by 28 (7 self)
 Add to MetaCart
We consider 2 × 2 block indefinite linear systems whose (2,2) block is zero. Such systems arise in many applications. We focus on techniques that change the linear systems in a way that may make it easier to solve them. In particular, two techniques based on modifying the (1,1) block are considered. The main part of the paper discusses an augmented Lagrangian approach, which is a technique that modifies the (1,1) block without changing the system size. The choice of the parameter involved, the spectrum of the linear system, and its condition number are discussed, and some analytical observations are provided. A technique of deating the (1,1) block is then introduced. Finally, numerical experiments which validate the analysis are presented.
Iterative Methods for Problems in Computational Fluid Dynamics
 ITERATIVE METHODS IN SCIENTIFIC COMPUTING
, 1996
"... We discuss iterative methods for solving the algebraic systems of equations arising from linearization and discretization of primitive variable formulations of the incompressible NavierStokes equations. Implicit discretization in time leads to a coupled but linear system of partial differential equ ..."
Abstract

Cited by 28 (5 self)
 Add to MetaCart
We discuss iterative methods for solving the algebraic systems of equations arising from linearization and discretization of primitive variable formulations of the incompressible NavierStokes equations. Implicit discretization in time leads to a coupled but linear system of partial differential equations at each time step, and discretization in space then produces a series of linear algebraic systems. We give an overview of commonly used time and space discretization techniques, and we discuss a variety of algorithmic strategies for solving the resulting systems of equations. The emphasis is on preconditioning techniques, which can be combined with Krylov subspace iterative methods. In many cases the solution of subsidiary problems such as the discrete convectiondiffusion equation and the discrete Stokes equations plays a crucial role. We examine iterative techniques for these problems and show how they can be integrated into effective solution algorithms for the NavierStokes equa...