Space-Time Reduced Basis Methods for Time-Periodic Partial Differential Equations

Steih, Kristina; Urban, Karsten

doi:10.3182/20120215-3-at-3016.00126

Cited by 15 publications

(14 citation statements)

References 10 publications

(17 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…(It is possible, alternatively, to consider a space-time RB approximation as well [11].) We define the common RB-quantities, namely the error e N (μ) :…”

Section: The Reduced Basis Methods (Rbm)mentioning

confidence: 99%

An improved error bound for reduced basis approximation of linear parabolic problems

2013

Self Cite

View full text Add to dashboard Cite

Abstract. We consider a space-time variational formulation for linear parabolic partial differential equations. We introduce an associated Petrov-Galerkin truth finite element discretization with favorable discrete inf-sup constant β δ , the inverse of which enters into error estimates: β δ is unity for the heat equation; β δ decreases only linearly in time for non-coercive (but asymptotically stable) convection operators. The latter in turn permits effective long-time a posteriori error bounds for reduced basis approximations, in sharp contrast to classical (pessimistic) exponentially growing energy estimates. The paper contains a full analysis and various extensions for the formulation introduced briefly by Urban and Patera (2012) as well as numerical results for a model reaction-convection-diffusion equation.

show abstract

“…(It is possible, alternatively, to consider a space-time RB approximation as well [11].) We define the common RB-quantities, namely the error e N (μ) :…”

Section: The Reduced Basis Methods (Rbm)mentioning

confidence: 99%

An improved error bound for reduced basis approximation of linear parabolic problems

2013

Self Cite

View full text Add to dashboard Cite

show abstract

“…The RB output is then given by s N = I (u N (t))dt(= I (ū N (t))dt). (It is possible, alternatively, to consider a space-time RB approximation as well [7].) It is then simple [5] We briefly comment on the computational implications of this space-time error bound.…”

Section: The Reduced Basis Methods (Rbm)mentioning

confidence: 99%

A New Error Bound for Reduced Basis Approximation of Parabolic Partial Differential Equations

Urban¹,

Patera²

2012

Self Cite

View full text Add to dashboard Cite

We consider a space-time variational formulation for linear parabolic partial differential equations. We introduce an associated Petrov-Galerkin truth finite element discretization with favorable discrete inf-sup constant β δ : β δ is unity for the heat equation; β δ grows only linearly in time for non-coercive (but asymptotically stable) convection operators. The latter in turn permits effective long-time a posteriori error bounds for reduced basis approximations, in sharp contrast to classical (pessimistic) exponentially growing energy estimates. RésuméNous considérons une formulation variationnelle espace-temps pour leséquations différentielles paraboliques linéaires. Nous y associons une discrétisation paréléments finis de Petrov-Galerkin pour laquelle la constante de stabilité inf-sup β δ possède des propriétés agréables : β δ est unité pour l'équation de la chaleur; β δ a une croissance seulement linéaire en temps pour des opérateurs de convection non-coercifs (mais asymptotiquement stables). Dans le cadre des approximations par bases réduites, cette dernière propriété permet d'obtenir des bornes efficaces pour l'erreur a posteriori en temps long, en net contraste avec les estimateurs d'erreur enénergie classiques (pessimistes) qui présentent une croissance exponentielle.

show abstract

“…A number of publications have considered model order reduction by projection onto a reduced space generated from space-adapted snapshots: Reduced basis methods with space-adapted snapshots are considered by [3,36] in the context of parametrized partial differential equations. The authors derive estimates of the error with respect to the infinite-dimensional truth solution.…”

Section: Introductionmentioning

confidence: 99%

POD model order reduction with space-adapted snapshots for incompressible flows

et al. 2019

View full text Add to dashboard Cite

We consider model order reduction based on proper orthogonal decomposition (POD) for unsteady incompressible Navier-Stokes problems, assuming that the snapshots are given by spatially adapted finite element solutions. We propose two approaches of deriving stable POD-Galerkin reduced-order models for this context. In the first approach, the pressure term and the continuity equation are eliminated by imposing a weak incompressibility constraint with respect to a pressure reference space. In the second approach, we derive an inf-sup stable velocity-pressure reduced-order model by enriching the velocity reduced space with supremizers computed on a velocity reference space. For problems with inhomogeneous Dirichlet conditions, we show how suitable lifting functions can be obtained from standard adaptive finite element computations. We provide a numerical comparison of the considered methods for a regularized lid-driven cavity problem.

show abstract

Space-Time Reduced Basis Methods for Time-Periodic Partial Differential Equations

Cited by 15 publications

References 10 publications

An improved error bound for reduced basis approximation of linear parabolic problems

An improved error bound for reduced basis approximation of linear parabolic problems

A New Error Bound for Reduced Basis Approximation of Parabolic Partial Differential Equations

POD model order reduction with space-adapted snapshots for incompressible flows

Contact Info

Product

Resources

About