Computational Aspects of Pseudospectra in Hydrodynamic Stability Analysis

Gerecht, Daniel; Rannacher, Rolf; Wollner, Winnifried

doi:10.1007/s00021-011-0085-7

Cited by 6 publications

(2 citation statements)

References 21 publications

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“…So far only few results are available concerning the relations between the pseudospectra of the discretized operator and those of the infinitedimensional operator. Convergence properties of the pseudospectrum under discretization have been studied for the linearized Navier-Stokes equation [9], for band-dominated bounded operators [18] and for Toeplitz operators [5]. Bögli and Siegl [3,4] prove local and global convergence of the pseudospectra of a sequence of linear operators which converge in a generalized resolvent sense.…”

Section: Introductionmentioning

confidence: 99%

Pseudospectrum Enclosures by Discretization

Frommer

Jacob

Vorberg

et al. 2021

Integr. Equ. Oper. Theory

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A new method to enclose the pseudospectrum via the numerical range of the inverse of a matrix or linear operator is presented. The method is applied to finite-dimensional discretizations of an operator on an infinite-dimensional Hilbert space, and convergence results for different approximation schemes are obtained, including finite element methods. We show that the pseudospectrum of the full operator is contained in an intersection of sets which are expressed in terms of the numerical ranges of shifted inverses of the approximating matrices. The results are illustrated by means of two examples: the advection–diffusion operator and the Hain–Lüst operator.

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Section: Introductionmentioning

confidence: 99%

Pseudospectrum Enclosures by Discretization

Frommer

Jacob

Vorberg

et al. 2021

Integr. Equ. Oper. Theory

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“…As a final comment, we recall that, due to the intrinsic complexity of the nondegeneracy conditions arising in the mathematical treatment of bifurcation, our analytic findings must be complemented or supported by numerical tests. For instance, the numerical computation in [17,18] illustrate the required nondegeneracy conditions given in [23]. We also mention a recent research line based on fully rigorous computer assisted proofs [2,3].…”

Section: Introductionmentioning

confidence: 99%

Equilibrium configuration of a rectangular obstacle immersed in a channel flow

Bonheure¹,

Galdi²,

Gazzola³

2020

Comptes Rendus. Mathématique

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Fluid flows around an obstacle generate vortices which, in turn, generate lift forces on the obstacle. Therefore, even in a perfectly symmetric framework equilibrium positions may be asymmetric. We show that this is not the case for a Poiseuille flow in an unbounded 2D channel, at least for small Reynolds number and flow rate. We consider both the cases of vertically moving obstacles and obstacles rotating around a fixed pin.

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On Bifurcating Time-Periodic Flow of a Navier-Stokes Liquid Past a Cylinder

Galdi

2016

Arch Rational Mech Anal

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We provide general sufficient conditions for branching out of a timeperiodic family of solutions from steady-state solutions to the twodimensional Navier-Stokes equations in the exterior of a cylinder. To this end, we first show that the problem can be formulated as a coupled elliptic-parabolic nonlinear system in appropriate function spaces. This is obtained by separating the time-independent averaged component of the velocity field from its "purely periodic" one. We then prove that time-periodic bifurcation occurs, provided the linearized time-independent operator of the parabolic problem possess a simple eigenvalue that crosses the imaginary axis when the Reynolds number passes through a (suitably defined) critical value. We also show that only supercritical or subcritical bifurcation may occur. Our approach is different and, we believe, more direct than those used by previous authors in similar, but distinct, context.

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Computational Aspects of Pseudospectra in Hydrodynamic Stability Analysis

Cited by 6 publications

References 21 publications

Pseudospectrum Enclosures by Discretization

Pseudospectrum Enclosures by Discretization

Equilibrium configuration of a rectangular obstacle immersed in a channel flow

On Bifurcating Time-Periodic Flow of a Navier-Stokes Liquid Past a Cylinder

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