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A fictitious domain approach to the direct numerical simulation of incompressible viscous flow past moving rigid bodies: application to particulate flow
 J. Comp. Phys
"... In this article we discuss a methodology that allows the direct numerical simulation of incompressible viscous fluid flow past moving rigid bodies. The simulation methods rest essentially on the combination of: (a) Lagrangemultiplierbased fictitious domain methods which allow the fluid flow compu ..."
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Cited by 114 (4 self)
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In this article we discuss a methodology that allows the direct numerical simulation of incompressible viscous fluid flow past moving rigid bodies. The simulation methods rest essentially on the combination of: (a) Lagrangemultiplierbased fictitious domain methods which allow the fluid flow computations to be done in a fixed flow region. (b) Finite element approximations of the Navier–Stokes equations occurring in the global model. (c) Time discretizations by operator splitting schemes in order to treat optimally the various operators present in the model. The above methodology is particularly well suited to the direct numerical simulation of particulate flow, such as the flow of mixtures of rigid solid particles and incompressible viscous fluids, possibly nonNewtonian. We conclude this article with the presentation of the results of various numerical experiments, including the simulation of store separation for rigid airfoils and of sedimentation and fluidization phenomena in two and three dimensions. c ° 2001 Academic Press Key Words: fictitious domain methods; finite element methods; distributed Lagrange multipliers; Navier–Stokes equations; particulate flow; liquid–solid mix
Performance and analysis of saddle point preconditioners for the discrete steadystate NavierStokes equations
 NUMER. MATH. (2002) 90: 665–688
, 2002
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An augmented Lagrangianbased approach to the Oseen problem
 SIAM J. SCI. COMPUT
, 2006
"... 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 mult ..."
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Cited by 68 (19 self)
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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.
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 ..."
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Cited by 61 (15 self)
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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.
Performance and accuracy of hardwareoriented native, emulated and mixedprecision solvers in FEM simulations
 International Journal of Parallel, Emergent and Distributed Systems
"... In a previous publication, we have examined the fundamental difference between computational precision and result accuracy in the context of the iterative solution of linear systems as they typically arise in the Finite Element discretization of Partial Differential Equations (PDEs) [1]. In particul ..."
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Cited by 54 (12 self)
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In a previous publication, we have examined the fundamental difference between computational precision and result accuracy in the context of the iterative solution of linear systems as they typically arise in the Finite Element discretization of Partial Differential Equations (PDEs) [1]. In particular, we evaluated mixed and emulatedprecision schemes on commodity graphics processors (GPUs), which at that time only supported computations in single precision. With the advent of graphics cards that natively provide double precision, this report updates our previous results. We demonstrate that with new coprocessor hardware supporting native double precision, such as NVIDIA’s G200 architecture, the situation does not change qualitatively for PDEs, and the previously introduced mixed precision schemes are still preferable to double precision alone. But the schemes achieve significant quantitative performance improvements with the more powerful hardware. In particular, we demonstrate that a Multigrid scheme can accurately solve a common test problem in Finite Element settings with one million unknowns in less than 0.1 seconds, which is truely outstanding performance. We support these conclusions by exploring the algorithmic design space enlarged by the availability of double precision directly in the hardware. 1 Introduction and
Algebraic flux correction I. Scalar conservation laws. Chapter 6 in the first edition of this book
, 2005
"... Abstract This chapter is concerned with the design of highresolution finite element schemes satisfying the discrete maximum principle. The presented algebraic flux correction paradigm is a generalization of the fluxcorrected transport (FCT) methodology. Given the standard Galerkin discretization ..."
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Cited by 45 (23 self)
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Abstract This chapter is concerned with the design of highresolution finite element schemes satisfying the discrete maximum principle. The presented algebraic flux correction paradigm is a generalization of the fluxcorrected transport (FCT) methodology. Given the standard Galerkin discretization of a scalar transport equation, we decompose the antidiffusive part of the discrete operator into numerical fluxes and limit these fluxes in a conservative way. The purpose of this manipulation is to make the antidiffusive term local extremum diminishing. The available limiting techniques include a family of implicit FCT schemes and a new linearitypreserving limiter which provides a unified treatment of stationary and timedependent problems. The use of Anderson acceleration makes it possible to design a simple and efficient quasiNewton solver for the constrained Galerkin scheme. We also present a linearized FCT method for computations with small time steps. The numerical behavior of the proposed algorithms is illustrated by a grid convergence study for convectiondominated transport problems and anisotropic diffusion equations. 1
Inversion of 3D electromagnetic data in frequency and time domain using an inexact allatonce approach
 Geophys
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Splitting methods based on algebraic factorization for fluidstructure interaction
 SIAM J. Scientific Computing
"... Abstract. We discuss in this paper the numerical approximation of fluidstructure interaction (FSI) problems dealing with strong addedmass effect. We propose new semiimplicit algorithms based on inexact blockLU factorization of the linear system obtained after the spacetime discretization and li ..."
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Cited by 42 (11 self)
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Abstract. We discuss in this paper the numerical approximation of fluidstructure interaction (FSI) problems dealing with strong addedmass effect. We propose new semiimplicit algorithms based on inexact blockLU factorization of the linear system obtained after the spacetime discretization and linearization of the FSI problem. As a result, the fluid velocity is computed separately from the coupled pressurestructure velocity system at each iteration, reducing the computational cost. We investigate explicitimplicit decomposition through algebraic splitting techniques originally designed for the FSI problem. This approach leads to two different families of methods which extend to FSI the algebraic pressure correction method and the Yosida method, two schemes that were previously adopted for pure fluid problems. Furthermore, we have considered the inexact factorization of the fluidstructure system as a preconditioner. The numerical properties of these methods have been tested on a model problem representing a bloodvessel system.
Modified augmented Lagrangian preconditioners for the incompressible Navier–Stokes equations
, 1002
"... We study different variants of the augmented Lagrangianbased block triangular preconditioner introduced by the first two authors in [SIAM J. Sci. Comput., 28 (2006), pp. 2095–2113]. The preconditioners are used to accelerate the convergence of the Generalized Minimal Residual (GMRES) method applied ..."
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Cited by 41 (10 self)
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We study different variants of the augmented Lagrangianbased block triangular preconditioner introduced by the first two authors in [SIAM J. Sci. Comput., 28 (2006), pp. 2095–2113]. The preconditioners are used to accelerate the convergence of the Generalized Minimal Residual (GMRES) method applied to various finite element and MAC discretizations of the Oseen problem in two and three space dimensions. Both steady and unsteady problems are considered. Numerical experiments show the effectiveness of the proposed preconditioners for a wide range of problem parameters. Implementation on parallel architectures is also considered. The augmented Lagrangianbased approach is further generalized to deal with linear systems from stabilized finite element discretizations. Copyright c ○ 2000 John Wiley & Sons, Ltd. key words: preconditioning; saddle point problems; Oseen problem; augmented Lagrangian method; Krylov subspace methods; parallel computing 1.
Flux correction tools for finite elements
 J. Comput. Phys
"... Peculiarities of flux correction in the finite element context are investigated. Criteria for positivity of the numerical solution are formulated, and the loworder transport operator is constructed from the discrete highorder operator by adding modulated dissipation so as to eliminate negative off ..."
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Cited by 38 (21 self)
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Peculiarities of flux correction in the finite element context are investigated. Criteria for positivity of the numerical solution are formulated, and the loworder transport operator is constructed from the discrete highorder operator by adding modulated dissipation so as to eliminate negative offdiagonal entries. The corresponding antidiffusive terms can be decomposed into a sum of genuine fluxes (rather than element contributions) which represent bilateral mass exchange between individual nodes. Thereby essentially onedimensional flux correction tools can be readily applied to multidimensional problems involving unstructured meshes. The proposed methodology guarantees mass conservation and makes it possible to design both explicit and implicit FCT schemes based on a unified limiting strategy. Numerical results for a number of benchmark problems illustrate the performance of the algorithm. Key Words: high resolution; finite elements; flux correction; positivity; mass conservation; unconditional stability 1