Analysis of Iterative Methods for the Steady and Unsteady Stokes Problem: Application to Spectral Element Discretizations

Maday, Yvon; Meiron, D. I.; Patera, Anthony T.; Rønquist, Einar M.

doi:10.1137/0914020

Cited by 112 publications

(92 citation statements)

References 26 publications

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“…(4.14), that a necessary condition for the stability of h/p methods, such as CG or DG, where both h and p refinement can be performed independently is dictated by: 17) where C h/p denotes a constant independent of the spatial discretisation. We also note that Maday et al [27] showed, in the context of the Uzawa algorithm when using high order C 0 schemes, that the inf-sup constant relates to the minimum eigenvalue of the pressure Schur complement of the discrete NS system, which is consistent with our analysis.…”

Section: Algebraic Systemsupporting

confidence: 91%

Stability of Projection Methods for Incompressible Flows Using High Order Pressure-Velocity Pairs of Same Degree: Continuous and Discontinuous Galerkin Formulations

Ferrer

Moxey

Willden

et al. 2014

Commun. comput. phys.

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Abstract. This paper presents limits for stability of projection type schemes when using high order pressure-velocity pairs of same degree. Two high order h/p variational methods encompassing continuous and discontinuous Galerkin formulations are used to explain previously observed lower limits on the time step for projection type schemes to be stable [18], when h-or p-refinement strategies are considered. In addition, the analysis included in this work shows that these stability limits do not depend only on the time step but on the product of the latter and the kinematic viscosity, which is of particular importance in the study of high Reynolds number flows. We show that high order methods prove advantageous in stabilising the simulations when small time steps and low kinematic viscosities are used.Drawing upon this analysis, we demonstrate how the effects of this instability can be reduced in the discontinuous scheme by introducing a stabilisation term into the global system. Finally, we show that these lower limits are compatible with CourantFriedrichs-Lewy (CFL) type restrictions, given that a sufficiently high polynomial order or a mall enough mesh spacing is selected.

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Section: Algebraic Systemsupporting

confidence: 91%

Stability of Projection Methods for Incompressible Flows Using High Order Pressure-Velocity Pairs of Same Degree: Continuous and Discontinuous Galerkin Formulations

Ferrer

Moxey

Willden

et al. 2014

Commun. comput. phys.

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“…First we orthogonalize the elements in X 0 N , followed by the elements in X e N . For the fully coupled system of Method 1, the corresponding Uzawa pressure operator can be constructed explicitly, and the smallest eigenvalue of this operator is directly connected to the inf-sup parameter β; see [14]. By computing numerically the minimum eigenvalue and the condition number of the Uzawa pressure operator, the numerical results indicate that these are constant for our problem and equal to 7.3×10 −2 and 14, respectively, independent of N (at least for the range of values we are dealing with).…”

Section: Numerical Resultsmentioning

confidence: 99%

A reduced basis element method for the steady Stokes problem

Løvgren¹,

Maday

Rønquist³

2006

ESAIM: M2AN

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Abstract. The reduced basis element method is a new approach for approximating the solution of problems described by partial differential equations. The method takes its roots in domain decomposition methods and reduced basis discretizations. The basic idea is to first decompose the computational domain into a series of subdomains that are deformations of a few reference domains (or generic computational parts). Associated with each reference domain are precomputed solutions corresponding to the same governing partial differential equation, but solved for different choices of deformations of the reference subdomains and mapped onto the reference shape. The approximation corresponding to a new shape is then taken to be a linear combination of the precomputed solutions, mapped from the generic computational part to the actual computational part. We extend earlier work in this direction to solve incompressible fluid flow problems governed by the steady Stokes equations. Particular focus is given to satisfying the inf-sup condition, to a posteriori error estimation, and to "gluing" the local solutions together in the multidomain case.Mathematics Subject Classification. 65C20, 65N15, 65N30, 65N35, 76D07, 93A30.

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“…Further details regarding this kind of Schur-complement approximation can be found in [16] for generalized Stokes systems, in [29,14] for steady and unsteady Stokes systems, and in [30] for general partial differential equations. For unsteady Stokes problems the above approximation is known as the Cahouet-Chabard preconditioner [16].…”

Section: −1mentioning

confidence: 99%

Efficient Solution of Large-Scale Saddle Point Systems Arising in Riccati-Based Boundary Feedback Stabilization of Incompressible Stokes Flow

Benner¹,

Saak²,

Stoll³

et al. 2013

SIAM J. Sci. Comput.

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Abstract. We investigate numerical methods for solving large-scale saddle point systems which arise during the feedback control of flow problems. We focus on the Stokes equations that describe instationary, incompressible flows for low and moderate Reynolds numbers. After a mixed finite element discretization [23] we get a differential-algebraic system of differential index two [45]. To reduce this index, we follow the analytic ideas of Raymond [34,35,36] coupled with the projection idea of Heinkenschloss et al. [22]. Avoiding this explicit projection leads to solving a series of largescale saddle point systems. In this paper we construct iterative methods to solve such saddle point systems by deriving efficient preconditioners based on the approaches of Wathen et al. [19,42]. In addition, the main results can be extended to the non-symmetric case of linearized Navier-Stokes equations. We conclude with numerical examples showcasing the performance of our preconditioned iterative saddle point solver.

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Analysis of Iterative Methods for the Steady and Unsteady Stokes Problem: Application to Spectral Element Discretizations

Cited by 112 publications

References 26 publications

Stability of Projection Methods for Incompressible Flows Using High Order Pressure-Velocity Pairs of Same Degree: Continuous and Discontinuous Galerkin Formulations

Stability of Projection Methods for Incompressible Flows Using High Order Pressure-Velocity Pairs of Same Degree: Continuous and Discontinuous Galerkin Formulations

A reduced basis element method for the steady Stokes problem

Efficient Solution of Large-Scale Saddle Point Systems Arising in Riccati-Based Boundary Feedback Stabilization of Incompressible Stokes Flow

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