(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

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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.
Original languageEnglish
Pages (from-to)A592-A621
Number of pages30
JournalSIAM journal on scientific computing
Volume41
Issue number1
DOIs
Publication statusPublished - 14 Feb 2019

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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. 2019 ; Vol. 41, No. 1. pp. A592-A621.
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Brunken, J, Smetana, K & Urban, K 2019, '(Parametrized) First Order Transport Equations: Realization of Optimally Stable Petrov-Galerkin Methods' SIAM journal on scientific computing, vol. 41, no. 1, pp. A592-A621. https://doi.org/10.1137/18M1176269

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

In: SIAM journal on scientific computing, Vol. 41, No. 1, 14.02.2019, p. A592-A621.

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

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Brunken J, Smetana K, Urban K. (Parametrized) First Order Transport Equations: Realization of Optimally Stable Petrov-Galerkin Methods. SIAM journal on scientific computing. 2019 Feb 14;41(1):A592-A621. https://doi.org/10.1137/18M1176269

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