Models of Incompressible Fluid Flow

Elman, Howard C.; Silvester, David J.; Wathen, Andrew J.

doi:10.1093/acprof:oso/9780199678792.003.0001

Cited by 79 publications

(212 citation statements)

References 10 publications

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“…We would like to point out that the criterion (9) is sometimes suggested in the literature on the Picard iteration, e.g. [8,5], but that an absolute termination criterion is suggested in [12]. There it is suggested to just "gain one digit", meaning to use a tolerance of 0.1.…”

Section: Application: Picard Iterationmentioning

confidence: 99%

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Termination criteria for inexact fixed‐point schemes

Birken

2015

Numerical Linear Algebra App

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We analyze inexact fixed point iterations where the generating function contains an inexact solve of an equation system to answer the question of how tolerances for the inner solves influence the iteration error of the outer fixed point iteration. Important applications are the Picard iteration and partitioned fluid structure interaction. We prove that the iteration converges irrespective of how accurate the inner systems are solved, provided that a specific relative termination criterion is employed, whereas standard relative and absolute criteria do not have this property. For the analysis, the iteration is modelled as a perturbed fixed point iteration and existing analysis is extended to the nested case x = F(S(x)).

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Section: Application: Picard Iterationmentioning

confidence: 99%

“…Strategies for choosing a termination criterion for the inner iteration are empirically discussed for example in [5,12].…”

Section: Introductionmentioning

confidence: 99%

Termination criteria for inexact fixed‐point schemes

Birken

2015

Numerical Linear Algebra App

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“…For standard saddle point problems in particular this is a method of choice, and is considered a state of the art when the problem is of very high dimension (see [10,25]). Such a solver proceeds by building up a Krylov subspace…”

Section: Krylov Solver and Preconditioningmentioning

confidence: 99%

Fast tensor product solvers for optimization problems with fractional differential equations as constraints

Dolgov

Pearson

Savostyanov

et al. 2016

Applied Mathematics and Computation

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Fractional differential equations have recently received much attention within computational mathematics and applied science, and their numerical treatment is an important research area as such equations pose substantial challenges to existing algorithms. An optimization problem with constraints given by fractional differential equations is considered, which in its discretized form leads to a high-dimensional tensor equation. To reduce the computation time and storage, the solution is sought in the tensor-train format. We compare three types of solution strategies that employ sophisticated iterative techniques using either preconditioned Krylov solvers or tailored alternating schemes. The competitiveness of these approaches is presented using several examples with constant and variable coefficients.

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“…Another interesting approach consists of an approximate block LU factorization of the system matrix. This type of preconditioners have been widely studied for the incompressible Navier-Stokes equations in the fluid mechanics community [22,21,18]. Recently, this approach has been applied to the full resistive MHD equations [9] (using the preconditioner as a solver in an operator splitting fashion) and to the 2D incompressible (reduced) resistive MHD formulation [19].…”

Section: Introductionmentioning

confidence: 99%

“…In order to define the Schur complement approximations, we have extended ideas from the techniques used for the incompressible Navier-Stokes equations to the inductionless MHD system. Moreover, a study of the exact Schur complement behavior and the effect of cancelling different terms in it has allowed us to propose an improved version of the Pressure-Convection-Diffusion (PCD) preconditioner (see [22]) where we introduce additional stabilization terms. The application of the MHD preconditioner to the thermally coupled problem is straightforward, since the coupling is in one direction only.…”

Section: Introductionmentioning

confidence: 99%

Block recursive LU preconditioners for the thermally coupled incompressible inductionless MHD problem

Badia

Martı́n

Planas

2014

Journal of Computational Physics

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The thermally coupled incompressible inductionless magnetohydrodynamics (MHD) problem models the flow of an electrically charged fluid under the influence of an external electromagnetic field with thermal coupling. This system of partial differential equations is strongly coupled and highly nonlinear for real cases of interest. Therefore, fully implicit time integration schemes are very desirable in order to capture the different physical scales of the problem at hand. However, solving the multiphysics linear systems of equations resulting from such algorithms is a very challenging task which requires efficient and scalable preconditioners. In this work, a new family of recursive block LU preconditioners is designed and tested for solving the thermally coupled inductionless MHD equations. These preconditioners are obtained after splitting the fully coupled matrix into one-physics problems for every variable (velocity, pressure, current density, electric potential and temperature) that can be optimally solved, e.g., using preconditioned domain decomposition algorithms. The main idea is to arrange the original matrix into an (arbitrary) 2 × 2 block matrix, and consider a LU preconditioner obtained by approximating the corresponding Schur complement. For every one of the diagonal blocks in the LU preconditioner, if it involves more than one type of unknown, we proceed the same way in a recursive fashion. This approach is stated in an abstract way, and can be straightforwardly applied to other multiphysics problems. Further, we precisely explain a flexible and general software design for the code implementation of this type of preconditioners.

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Models of Incompressible Fluid Flow

Abstract: This preliminary chapter gives a brief introduction to the equations studied in depth in the book, as they are used to model incompressible fluids.

Cited by 79 publications

References 10 publications

Termination criteria for inexact fixed‐point schemes

Termination criteria for inexact fixed‐point schemes

Fast tensor product solvers for optimization problems with fractional differential equations as constraints

Block recursive LU preconditioners for the thermally coupled incompressible inductionless MHD problem

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