(Parametrized) First Order Transport Equations: Realization of Optimally Stable Petrov-Galerkin Methods

Julia Brunken, Kathrin Smetana, Karsten Urban

Research output: Contribution to journal › Article › Academic › peer-review

Abstract

We consider ultraweak variational formulations for (parametrized) linear first order transport equations in time and/or space. Computationally feasible pairs of optimally stable trial and test spaces are presented, starting with a suitable test space and defining an optimal trial space by the application of the adjoint operator. As a result, the inf-sup constant is one in the continuous as well as in the discrete case and the computational realization is therefore easy. In particular, regarding the latter, we avoid a stabilization loop within the greedy algorithm when constructing reduced models within the framework of reduced basis methods. Several numerical experiments demonstrate the good performance of the new method.
LanguageEnglish
JournalSIAM journal on scientific computing
Publication statusAccepted/In press - 2018

Fingerprint

Petrov-Galerkin Method
Galerkin methods
Transport Equation
Stabilization
First-order
Experiments
Reduced Basis Methods
Adjoint Operator
Reduced Model
Variational Formulation
Greedy Algorithm
Numerical Experiment
Demonstrate

Keywords

  • Linear transport equation
  • inf-sup stability
  • reduced basis methods

Cite this

Brunken, Julia ; Smetana, Kathrin ; Urban, Karsten. / (Parametrized) First Order Transport Equations: Realization of Optimally Stable Petrov-Galerkin Methods. In: SIAM journal on scientific computing. 2018.
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Brunken, J, Smetana, K & Urban, K 2018, '(Parametrized) First Order Transport Equations: Realization of Optimally Stable Petrov-Galerkin Methods' SIAM journal on scientific computing.

(Parametrized) First Order Transport Equations: Realization of Optimally Stable Petrov-Galerkin Methods. / Brunken, Julia; Smetana, Kathrin ; Urban, Karsten.

In: SIAM journal on scientific computing, 2018.

Research output: Contribution to journal › Article › Academic › peer-review

TY - JOUR

T1 - (Parametrized) First Order Transport Equations: Realization of Optimally Stable Petrov-Galerkin Methods

AU - Brunken, Julia

AU - Smetana, Kathrin

AU - Urban, Karsten

PY - 2018

Y1 - 2018

N2 - We consider ultraweak variational formulations for (parametrized) linear first order transport equations in time and/or space. Computationally feasible pairs of optimally stable trial and test spaces are presented, starting with a suitable test space and defining an optimal trial space by the application of the adjoint operator. As a result, the inf-sup constant is one in the continuous as well as in the discrete case and the computational realization is therefore easy. In particular, regarding the latter, we avoid a stabilization loop within the greedy algorithm when constructing reduced models within the framework of reduced basis methods. Several numerical experiments demonstrate the good performance of the new method.

AB - We consider ultraweak variational formulations for (parametrized) linear first order transport equations in time and/or space. Computationally feasible pairs of optimally stable trial and test spaces are presented, starting with a suitable test space and defining an optimal trial space by the application of the adjoint operator. As a result, the inf-sup constant is one in the continuous as well as in the discrete case and the computational realization is therefore easy. In particular, regarding the latter, we avoid a stabilization loop within the greedy algorithm when constructing reduced models within the framework of reduced basis methods. Several numerical experiments demonstrate the good performance of the new method.

KW - Linear transport equation

KW - inf-sup stability

KW - reduced basis methods

M3 - Article

JO - SIAM journal on scientific computing

T2 - SIAM journal on scientific computing

JF - SIAM journal on scientific computing

SN - 1064-8275

ER -

Brunken J, Smetana K, Urban K. (Parametrized) First Order Transport Equations: Realization of Optimally Stable Petrov-Galerkin Methods. SIAM journal on scientific computing. 2018.

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