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A Dimensional Split Preconditioner for Stokes and Linearized Navier–Stokes Equations
, 2010
"... In this paper we introduce a new preconditioner for linear systems of saddle point type arising from the numerical solution of the Navier– Stokes equations. Our approach is based on a dimensional splitting of the problem along the components of the velocity field, resulting in a convergent fixedpoi ..."
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In this paper we introduce a new preconditioner for linear systems of saddle point type arising from the numerical solution of the Navier– Stokes equations. Our approach is based on a dimensional splitting of the problem along the components of the velocity field, resulting in a convergent fixedpoint iteration. The basic iteration is accelerated by a Krylov subspace method like restarted GMRES. The corresponding preconditioner requires at each iteration the solution of a set of discrete scalar elliptic equations, one for each component of the velocity field. Numerical experiments illustrating the convergence behavior for different finite element discretizations of Stokes and Oseen problems are included. Key words. saddle point problems, matrix splittings, iterative methods, preconditioning, Stokes problem, Oseen problem, stretched grids
ANALYSIS OF AUGMENTED LAGRANGIANBASED PRECONDITIONERS FOR THE STEADY INCOMPRESSIBLE NAVIERSTOKES EQUATIONS
"... Abstract. We analyze a class of modified augmented Lagrangianbased preconditioners for both stable and stabilized finite element discretizations of the steady incompressible Navier–Stokes equations. We study the eigenvalues of the preconditioned matrices obtained from Picard linearization, and we d ..."
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Abstract. We analyze a class of modified augmented Lagrangianbased preconditioners for both stable and stabilized finite element discretizations of the steady incompressible Navier–Stokes equations. We study the eigenvalues of the preconditioned matrices obtained from Picard linearization, and we devise a simple and effective method for the choice of the augmentation parameter γ based on Fourier analysis. Numerical experiments on a wide range of model problems demonstrate the robustness of these preconditioners, yielding fast convergence independent of mesh size and only mildly dependent on viscosity on both uniform and stretched grids. Good results are also obtained on linear systems arising from Newton linearization. We also show that performing inexact preconditioner solves with an algebraic multigrid algorithm results in excellent scalability. Comparisons of the modified augmented Lagrangian preconditioners with other stateoftheart techniques show the competitiveness of our approach.
ILU preconditioners for nonsymmetric saddle point matrices with application to the incompressible Navier–Stokes equations
 SIAM Journal on Scientific Computing
, 2015
"... Abstract. Motivated by the numerical solution of the linearized incompressible Navier–Stokes equations, we study threshold incomplete LU factorizations for nonsymmetric saddlepoint matrices. The resulting preconditioners are used to accelerate the convergence of a Krylov subspace method applied to ..."
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Abstract. Motivated by the numerical solution of the linearized incompressible Navier–Stokes equations, we study threshold incomplete LU factorizations for nonsymmetric saddlepoint matrices. The resulting preconditioners are used to accelerate the convergence of a Krylov subspace method applied to finite element discretizations of fluid dynamics problems in three space dimensions. The paper presents and examines an extension for nonsymmetric matrices of the Tismenetsky–Kaporin incomplete factorization. It is shown that in numerically challenging cases of higher Reynolds number flows one benefits from using this twoparameter modification of a standard threshold ILU preconditioner. The performance of the ILU preconditioners is studied numerically for a wide range of flow and discretization parameters, and the efficiency of the approach is shown if threshold parameters are chosen suitably. The practical utility of the method is further demonstrated for the haemodynamic problem of simulating blood flow in a right coronary artery of a real patient.
Purely algebraic domain decomposition methods for the incompressible NavierStokes equations. ArXiv eprints
, 2011
"... In the context of non overlapping domain decomposition methods, several algebraic approximations of the DirichlettoNeumann (DtN) map are proposed in [F. X. Roux, et. al.Algebraic approximation of DirichlettoNeumann maps for the equations of linear elasticity, Comput. Methods Appl. Mech. Engrg., ..."
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In the context of non overlapping domain decomposition methods, several algebraic approximations of the DirichlettoNeumann (DtN) map are proposed in [F. X. Roux, et. al.Algebraic approximation of DirichlettoNeumann maps for the equations of linear elasticity, Comput. Methods Appl. Mech. Engrg., 195, 2006, 37423759]. For the case of non overlapping domains, approximation to the DtN are analogous to the approximation of the Schur complements in the incomplete multilevel block factorization. In this work, several original and purely algebraic (based on graph of the matrix) domain decomposition techniques are investigated for steady state incompressible NavierStokes equation defined on uniform and stretched grid for low viscosity. Moreover, the methods proposed are highly parallel during both setup and application phase. Spectral and numerical analysis of the methods are also presented. 1
Parameter estimates for the Relaxed Dimensional Factorization preconditioner and application to hemodynamics
"... We present new results on the Relaxed Dimensional Factorization (RDF) preconditioner for solving saddle point problems from incompressible flow simulations, first introduced in [1]. This method contains a parameter α> 0, to be chosen by the user. Previous works provided an estimate of α in the 2D ..."
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We present new results on the Relaxed Dimensional Factorization (RDF) preconditioner for solving saddle point problems from incompressible flow simulations, first introduced in [1]. This method contains a parameter α> 0, to be chosen by the user. Previous works provided an estimate of α in the 2D case using Local Fourier Analysis. Novel algebraic estimation techniques for finding a suitable value of the RDF parameter in both the 2D and the 3D case with arbitrary geometries are proposed. These techniques are tested on a variety of discrete saddle point problems arising from the approximation of the Navier–Stokes equations using a MarkerandCell scheme and a finite element one. We also show results for a largescale problem relevant for hemodynamics simulation that we solve in parallel using up to 8196 cores.
PARAMETER ESTIMATES FOR THE RELAXED DIMENSIONAL FACTORIZATION PRECONDITIONER AND APPLICATION TO HEMODYNAMICS∗
"... Abstract. We present new results on the Relaxed Dimensional Factorization (RDF) preconditioner for solving saddle point problems, first introduced in [5]. This method contains a parameter α> 0, to be chosen by the user. Previous works provided an estimate of α in the 2D case using Local Fourier ..."
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Abstract. We present new results on the Relaxed Dimensional Factorization (RDF) preconditioner for solving saddle point problems, first introduced in [5]. This method contains a parameter α> 0, to be chosen by the user. Previous works provided an estimate of α in the 2D case using Local Fourier Analysis. Novel algebraic estimation techniques for finding a suitable value of the RDF parameter in both the 2D and the 3D case with arbitrary geometries are proposed. These techniques are tested on a variety of discrete saddle point problems arising from the approximation of the Navier–Stokes equations using a MarkerandCell scheme and a finite element one. We also show results for a largescale problem relevant for hemodynamics simulation that we solve in parallel using up to 8196 cores. Key words. scalable parallel preconditioners, finite element method, high performance computing, Navier–Stokes equations, hemodynamics applications, dimensional splitting preconditioner,