Volume 50, Number 1, January-February 2016
|Page(s)||163 - 185|
|Published online||01 December 2015|
A minimum-residual mixed reduced basis method: Exact residual certification and simultaneous finite-element reduced-basis refinement
Department of Mechanical Engineering, Massachusetts Institute of
Technology ; 77 Massachusetts Ave, Rm. 3-237, Cambridge, MA
Received: 18 August 2014
Revised: 4 February 2015
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.
Mathematics Subject Classification: 65N15 / 65N30 / 65N35
Key words: Minimum-residual mixed method / reduced basis method / a posteriori error bounds / offline-online decomposition / adaptivity
© EDP Sciences, SMAI, 2015
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