We present a reduced basis method for parametrized partial differential equations certified by a dual-norm bound of the residual computed not in the typical finite-element “truth” space but rather in an infinite-dimensional function space. The bound builds on a finite element method and an associated reduced-basis approximation derived from a minimum-residual mixed formulation. The offline stage combines a spatial mesh adaptation for finite elements and a greedy parameter sampling strategy for reduced bases to yield a reliable online system in an efficient manner; the online stage provides the solution and the associated dual-norm bound of the residual for any parameter value in complexity independent of the finite element resolution. We assess the effectiveness of the approach for a parametrized reaction-diffusion equation and a parametrized advection-diffusion equation with a corner singularity; not only does the residual bound provide reliable certificates for the solutions, the associated mesh adaptivity significantly reduces the offline computational cost for the reduced-basis generation and the greedy parameter sampling ensures quasi-optimal online complexity.

DOI: 10.1051/m2an/2015039

Keywords: Minimum-residual mixed method, reduced basis method, a posteriori error bounds, offline-online decomposition, adaptivity

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@article{M2AN_2016__50_1_163_0, author = {Yano, Masayuki}, title = {A minimum-residual mixed reduced basis method: {Exact} residual certification and simultaneous finite-element reduced-basis refinement}, journal = {ESAIM: Mathematical Modelling and Numerical Analysis }, pages = {163--185}, publisher = {EDP-Sciences}, volume = {50}, number = {1}, year = {2016}, doi = {10.1051/m2an/2015039}, zbl = {1335.65095}, mrnumber = {3460105}, language = {en}, url = {http://www.numdam.org/articles/10.1051/m2an/2015039/} }

TY - JOUR AU - Yano, Masayuki TI - A minimum-residual mixed reduced basis method: Exact residual certification and simultaneous finite-element reduced-basis refinement JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2016 SP - 163 EP - 185 VL - 50 IS - 1 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2015039/ DO - 10.1051/m2an/2015039 LA - en ID - M2AN_2016__50_1_163_0 ER -

%0 Journal Article %A Yano, Masayuki %T A minimum-residual mixed reduced basis method: Exact residual certification and simultaneous finite-element reduced-basis refinement %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2016 %P 163-185 %V 50 %N 1 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2015039/ %R 10.1051/m2an/2015039 %G en %F M2AN_2016__50_1_163_0

Yano, Masayuki. A minimum-residual mixed reduced basis method: Exact residual certification and simultaneous finite-element reduced-basis refinement. ESAIM: Mathematical Modelling and Numerical Analysis , Volume 50 (2016) no. 1, pp. 163-185. doi : 10.1051/m2an/2015039. http://www.numdam.org/articles/10.1051/m2an/2015039/

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