Geodesic
A minimum-residual mixed reduced basis method: Exact residual certification and simultaneous finite-element reduced-basis refinement
Yano, Masayuki1
1 Department of Mechanical Engineering, Massachusetts Institute of Technology; 77 Massachusetts Ave, Rm. 3-237, Cambridge, MA 02139, United States
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 50 (2016) no. 1, pp. 163-185

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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.

Reçu le :
DOI : 10.1051/m2an/2015039
Classification : 65N15, 65N30, 65N35
Keywords: Minimum-residual mixed method, reduced basis method, a posteriori error bounds, offline-online decomposition, adaptivity

Yano, Masayuki 1

1 Department of Mechanical Engineering, Massachusetts Institute of Technology; 77 Massachusetts Ave, Rm. 3-237, Cambridge, MA 02139, United States 
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     title = {A minimum-residual mixed reduced basis method: {Exact} residual certification and simultaneous finite-element reduced-basis refinement},
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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 , Tome 50 (2016) no. 1, pp. 163-185. doi: 10.1051/m2an/2015039

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