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Galerkin approximationwith proper orthogonal decomposition :new error estimates and illustrative examples

Published online by Cambridge University Press:  03 February 2012

Dominique Chapelle ,
Asven Gariah  and
Jacques Sainte-Marie
Dominique Chapelle
Affiliation:
Inria Rocquencourt Laboratoire Saint-Venant Domaine de Voluceau Rocquencourt, B.P. 105, 78153 Le Chesnay, France. [email protected]
Asven Gariah
Affiliation:
Inria Rocquencourt Laboratoire Saint-Venant Domaine de Voluceau Rocquencourt, B.P. 105, 78153 Le Chesnay, France. [email protected]
Jacques Sainte-Marie
Affiliation:
Inria Rocquencourt Laboratoire Saint-Venant Domaine de Voluceau Rocquencourt, B.P. 105, 78153 Le Chesnay, France. [email protected] CETMEF, 2 boulevard Gambetta, 60200 Compiègne, France
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Abstract

We propose a numerical analysis of proper orthogonal decomposition (POD) model reductions in which a priori error estimates are expressed in terms of the projection errors that are controlled in the construction of POD bases. These error estimates are derived for generic parabolic evolution PDEs, including with non-linear Lipschitz right-hand sides, and for wave-like equations. A specific projection continuity norm appears in the estimates and – whereas a general uniform continuity bound seems out of reach – we prove that such a bound holds in a variety of Galerkin bases choices. Furthermore, we directly numerically assess this bound – and the effectiveness of the POD approach altogether – for test problems of the type considered in the numerical analysis, and also for more complex equations. Namely, the numerical assessment includes a parabolic equation with super-linear reaction terms, inspired from the FitzHugh-Nagumo electrophysiology model, and a 3D biomechanical heart model. This shows that the effectiveness established for the simpler models is also achieved in the reduced-order simulation of these highly complex systems.

Keywords

PODGalerkin approximation error estimatesnon-linear parabolic problemscardiac models
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 46 , Issue 4 , July 2012 , pp. 731 - 757
Copyright
© EDP Sciences, SMAI, 2012

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