Finite dimensional approximation of nonlinear problems

Brezzi, Franco; Rappaz, Jacques; Raviart, Pierre-Arnaud

doi:10.1007/bf01395805

Cited by 211 publications

(226 citation statements)

References 11 publications

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“…on the FE method (see e.g. [21,27]). Nevertheless, bases constructed using the greedy algorithm provide reliable approximations also in the case of bifurcation points included in the parameter space; for instance, ROMs have been used to track particular solution branches past the bifurcation point, see e.g.…”

Section: "Divide and Conquer Whenever Possible"mentioning

confidence: 99%

“…In the Navier-Stokes case we can instead obtain a rigorous a posteriori error estimation by relying on the so-called Brezzi-Rappaz-Raviart (BRR) theory [21,27], which is useful for the analysis of a wider class of nonlinear equations. We require some slight modifications with respect to the linear preliminaries, even if also for the Navier-Stokes problem the a posteriori error estimation takes advantage of the dual norm of residuals and of an effective lower bound of a suitable (parametric) stability factor, given in this case by the Babuška inf-sup constant referred not to the global Navier-Stokes operator…”

Section: Certification Of Roms For the Steady Navier-stokes Equationsmentioning

confidence: 99%

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Model Order Reduction in Fluid Dynamics: Challenges and Perspectives

Lassila

Manzoni

Quarteroni

et al. 2014

Reduced Order Methods for Modeling and Computational Reduction

164

179

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This chapter reviews techniques of model reduction of fluid dynamics systems. Fluid systems are known to be difficult to reduce efficiently due to several reasons. First of all, they exhibit strong nonlinearities -which are mainly related either to nonlinear convection terms and/or some geometric variability -that often cannot be treated by simple linearization. Additional difficulties arise when attempting model reduction of unsteady flows, especially when long-term transient behavior needs to be accurately predicted using reduced order models and more complex features, such as turbulence or multiphysics phenomena, have to be taken into consideration. We first discuss some general principles that apply to many parametric model order reduction problems, then we apply them on steady and unsteady viscous flows modelled by the incompressible Navier-Stokes equations. We address questions of inf-sup stability, certification through error estimation, computational issues and -in the unsteady case -long-time stability of the reduced model. Moreover, we provide an extensive list of literature references.

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Section: "Divide and Conquer Whenever Possible"mentioning

confidence: 99%

Section: Certification Of Roms For the Steady Navier-stokes Equationsmentioning

confidence: 99%

Model Order Reduction in Fluid Dynamics: Challenges and Perspectives

Lassila

Manzoni

Quarteroni

et al. 2014

Reduced Order Methods for Modeling and Computational Reduction

164

179

View full text Add to dashboard Cite

show abstract

“…Assume that p satisfies the zero mean constraint, ∫ Ω p dx = 0. The objective of this optimal control problem is to seek a state variables u and p, and the control f which minimize the L 2 -norm distances between u and u d and satisfy (2). The second term in (1) is added as a limiting the cost of control and the positive penalty parameter β can be used to change the relative importance of the two terms appearing in the definition of the functional.…”

Section: Introductionmentioning

confidence: 99%

Convergence of the Newton's Method for an Optimal Control Problems for Navier-Stokes Equations

Choi¹,

Kim²,

Lee³

2011

Bulletin of the Korean Mathematical Society

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Abstract. We consider the Newton's method for an direct solver of the optimal control problems of the Navier-Stokes equations. We show that the finite element solutions of the optimal control problem for Stokes equations may be chosen as the initial guess for the quadratic convergence of Newton's algorithm applied to the optimal control problem for the Navier-Stokes equations provided there are sufficiently small mesh size h and the moderate Reynold's number.

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“…Some a priori estimates of this type have been given recently in [16], [17], [18]. But, as in the linear case, we require, of course, computable and reliable a posteriori estimators which can then form the basis of adaptive procedures for computing with a prescribed accuracy the desired segments of solution paths on the equilibrium surface.…”

Section: Introductionmentioning

confidence: 99%