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Volume 28, Issue 1
Reduced Basis Method for Parametrized Elliptic Advection-Reaction Problems

Luca Dedè

DOI: 10.4208/jcm.2009.09-m3015

J. Comp. Math., 28 (2010), pp. 122-148.

Published online: 2010-02

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  • Abstract

In this work we consider the Reduced Basis method for the solution of parametrized advection-reaction partial differential equations. For the generation of the basis we adopt a stabilized finite element method and we define the Reduced Basis method in the "primal-dual" formulation for this stabilized problem. We provide a priori Reduced Basis error estimates and we discuss the effects of the finite element approximation on the Reduced Basis error. We propose an adaptive algorithm, based on the a posteriori Reduced Basis error estimate, for the selection of the sample sets upon which the basis are built; the idea leading this algorithm is the minimization of the computational costs associated with the solution of the Reduced Basis problem. Numerical tests demonstrate the efficiency, in terms of computational costs, of the "primal-dual" Reduced Basis approach with respect to an "only primal" one.

  • Keywords

Parametrized advection–reaction partial differential equations, Reduced Basis method, “primal–dual” reduced basis approach, Stabilized finite element method, a posteriori error estimation.

  • AMS Subject Headings

35J25, 35L50, 65N15, 65N30, 76R99.

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{JCM-28-122, author = {Luca Dedè}, title = {Reduced Basis Method for Parametrized Elliptic Advection-Reaction Problems}, journal = {Journal of Computational Mathematics}, year = {2010}, volume = {28}, number = {1}, pages = {122--148}, abstract = {

In this work we consider the Reduced Basis method for the solution of parametrized advection-reaction partial differential equations. For the generation of the basis we adopt a stabilized finite element method and we define the Reduced Basis method in the "primal-dual" formulation for this stabilized problem. We provide a priori Reduced Basis error estimates and we discuss the effects of the finite element approximation on the Reduced Basis error. We propose an adaptive algorithm, based on the a posteriori Reduced Basis error estimate, for the selection of the sample sets upon which the basis are built; the idea leading this algorithm is the minimization of the computational costs associated with the solution of the Reduced Basis problem. Numerical tests demonstrate the efficiency, in terms of computational costs, of the "primal-dual" Reduced Basis approach with respect to an "only primal" one.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.2009.09-m3015}, url = {http://global-sci.org/intro/article_detail/jcm/8511.html} }
TY - JOUR T1 - Reduced Basis Method for Parametrized Elliptic Advection-Reaction Problems AU - Luca Dedè JO - Journal of Computational Mathematics VL - 1 SP - 122 EP - 148 PY - 2010 DA - 2010/02 SN - 28 DO - http://doi.org/10.4208/jcm.2009.09-m3015 UR - https://global-sci.org/intro/article_detail/jcm/8511.html KW - Parametrized advection–reaction partial differential equations, Reduced Basis method, “primal–dual” reduced basis approach, Stabilized finite element method, a posteriori error estimation. AB -

In this work we consider the Reduced Basis method for the solution of parametrized advection-reaction partial differential equations. For the generation of the basis we adopt a stabilized finite element method and we define the Reduced Basis method in the "primal-dual" formulation for this stabilized problem. We provide a priori Reduced Basis error estimates and we discuss the effects of the finite element approximation on the Reduced Basis error. We propose an adaptive algorithm, based on the a posteriori Reduced Basis error estimate, for the selection of the sample sets upon which the basis are built; the idea leading this algorithm is the minimization of the computational costs associated with the solution of the Reduced Basis problem. Numerical tests demonstrate the efficiency, in terms of computational costs, of the "primal-dual" Reduced Basis approach with respect to an "only primal" one.

Luca Dedè. (2010). Reduced Basis Method for Parametrized Elliptic Advection-Reaction Problems. Journal of Computational Mathematics. 28 (1). 122-148. doi:10.4208/jcm.2009.09-m3015
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