Block Preconditioners Based on Approximate Commutators

Elman, Howard C.; Howle, Victoria E.; Shadid, John N.; Shuttleworth, Robert; Tuminaro, Raymond S.

doi:10.1137/040608817

Cited by 155 publications

(153 citation statements)

References 16 publications

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“…Therefore, Q −1 u is replaced witĥ Q −1 u , the inverse of the diagonal of the velocity mass matrix. According to the literature ( [13], [11]), the convergence of the LSC and PCD preconditioners is independent of the grid size, and only mildly dependent on the Reynolds number. The preconditioners have been tested for the driven cavity and backward facing step problem using Q2-Q1 and Q2-P1 elements.…”

Section: Preconditioning Strategies Based On Block Factorizationmentioning

confidence: 99%

“…From the eigenvalues, it is evident that convergence with the preconditioners based on formulation (5) depends strongly on the approximation to the Schur complement matrix. In ( [11], [10], [13]) one can find examples of such preconditioners.…”

Section: Preconditioning Strategies Based On Block Factorizationmentioning

confidence: 99%

“…Elman et al [11] created a matrix F p that minimizes the commutator (7). This can be achieved by solving a least squares problem.…”

Section: Preconditioning Strategies Based On Block Factorizationmentioning

confidence: 99%

“…The preconditioners are based on the knowledge of the pressure and velocity matrices. Examples of such preconditioners can be found in [10][11][12][13][14][15][16]. The goal of all these preconditioners is to get convergence independent of the Reynolds number and grid size.…”

Section: Introductionmentioning

confidence: 99%

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SIMPLE‐type preconditioners for the Oseen problem

Rehman

Vuik

Segal

2008

Numerical Methods in Fluids

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In this paper, the steady incompressible Navier–Stokes equations are discretized by the finite element method. The resulting systems of equations are solved by preconditioned Krylov subspace methods. Some new preconditioning strategies, both algebraic and problem dependent are discussed. We emphasize on the approximation of the Schur complement as used in semi implicit method for pressure‐linked equations‐type preconditioners. In the usual formulation, the Schur complement matrix and updates use scaling with the diagonal of the convection–diffusion matrix. We propose a variant of the SIMPLER preconditioner. Instead of using the diagonal of the convection–diffusion matrix, we scale the Schur complement and updates with the diagonal of the velocity mass matrix. This variant is called modified SIMPLER (MSIMPLER). With the new approximation, we observe a drastic improvement in convergence for large problems. MSIMPLER shows better convergence than the well‐known least‐squares commutator preconditioner which is also based on the diagonal of the velocity mass matrix. Copyright © 2008 John Wiley & Sons, Ltd.

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Section: Preconditioning Strategies Based On Block Factorizationmentioning

confidence: 99%

Section: Preconditioning Strategies Based On Block Factorizationmentioning

confidence: 99%

“…Elman et al [11] created a matrix F p that minimizes the commutator (7). This can be achieved by solving a least squares problem.…”

Section: Preconditioning Strategies Based On Block Factorizationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

SIMPLE‐type preconditioners for the Oseen problem

Rehman

Vuik

Segal

2008

Numerical Methods in Fluids

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“…whereŜ is approximated by a least squares commutator approach [10]. The approximation is based on commutator of the convection diffusion operator (on the velocity and pressure space) with the gradient operator.…”

Section: Lsc Preconditionermentioning

confidence: 99%

Block Preconditioners for the Incompressible Stokes Problem

Rehman

Vuik

Segal

2010

Large-Scale Scientific Computing

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Abstract. This paper discusses the solution of the Stokes problem using block preconditioned iterative methods. Block preconditioners are based on the block factorization of the discretized problem. We focus on two specific types: SIMPLE-type preconditioners and the LSC preconditioner. Both methods use scaling to improve their performance. We test convergence of GCR in combination with these preconditioners both for a constant and a non-constant viscosity Stokes problem.

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Immersed boundary model of aortic heart valve dynamics with physiological driving and loading conditions

Griffith

2011

Numer Methods Biomed Eng

154

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The immersed boundary (IB) method is a mathematical and numerical framework for problems of fluid–structure interaction, treating the particular case in which an elastic structure is immersed in a viscous incompressible fluid. The IB approach to such problems is to describe the elasticity of the immersed structure in Lagrangian form, and to describe the momentum, viscosity, and incompressibility of the coupled fluid–structure system in Eulerian form. Interaction between Lagrangian and Eulerian variables is mediated by integral equations with Dirac delta function kernels. The IB method provides a unified formulation for fluid–structure interaction models involving both thin elastic boundaries and also thick viscoelastic bodies. In this work, we describe the application of an adaptive, staggered-grid version of the IB method to the three-dimensional simulation of the fluid dynamics of the aortic heart valve. Our model describes the thin leaflets of the aortic valve as immersed elastic boundaries, and describes the wall of the aortic root as a thick, semi-rigid elastic structure. A physiological left-ventricular pressure waveform is used to drive flow through the model valve, and dynamic pressure loading conditions are provided by a reduced (zero-dimensional) circulation model that has been fit to clinical data. We use this model and method to simulate aortic valve dynamics over multiple cardiac cycles. The model is shown to approach rapidly a periodic steady state in which physiological cardiac output is obtained at physiological pressures. These realistic flow rates are not specified in the model, however. Instead, they emerge from the fluid–structure interaction simulation.

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Block Preconditioners Based on Approximate Commutators

Abstract: This paper introduces a strategy for automatically generating a block preconditioner for solving the incompressible Navier-Stokes equations. We consider the "pressure convectiondiffusion preconditioners" proposed by Kay, Loghin, and Wathen [SIAM

Cited by 155 publications

References 16 publications

SIMPLE‐type preconditioners for the Oseen problem

SIMPLE‐type preconditioners for the Oseen problem

Block Preconditioners for the Incompressible Stokes Problem

Immersed boundary model of aortic heart valve dynamics with physiological driving and loading conditions

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