Uniformly exponentially stable approximations for a class of damped systems

Ervedoza S, Zuazua E (2009)

Publication Type: Journal article

Publication year: 2009

Journal

Journal De Mathematiques Pures Et Appliquees Elsevier

Book Volume: 91

Pages Range: 20-48

Journal Issue: 1

DOI: 10.1016/j.matpur.2008.09.002

Abstract

We consider time semi-discrete approximations of a class of exponentially stable infinite-dimensional systems modeling, for instance, damped vibrations. It has recently been proved that for time semi-discrete systems, due to high frequency spurious components, the exponential decay property may be lost as the time step tends to zero. We prove that adding a suitable numerical viscosity term in the numerical scheme, one obtains approximations that are uniformly exponentially stable. This result is then combined with previous ones on space semi -di scretizati ons to derive similar results on fully-discrete approximation schemes. Our method is mainly based on a decoupling argument of low and high frequencies, the low frequency observability property for time semi-discrete approximations of conservative linear systems and the dissipativity of the numerical viscosity on the high frequency components. Our methods also allow to deal directly with stabilization properties of fully discrete approximation schemes without numerical viscosity, under a Suitable CFL type condition on the time and space discretization parameters. (c) 2008 Elsevier Masson SAS. All tights reserved.

Authors with CRIS profile

Enrique Zuazua Iriondo

Involved external institutions

Versailles Saint-Quentin-en-Yvelines University / Université de Versailles Saint-Quentin-en-Yvelines (UVSQ) FR France (FR) Basque Center of Applied Mathematics (BCAM) ES Spain (ES)

How to cite

APA:

Ervedoza, S., & Zuazua, E. (2009). Uniformly exponentially stable approximations for a class of damped systems. Journal De Mathematiques Pures Et Appliquees, 91(1), 20-48. https://dx.doi.org/10.1016/j.matpur.2008.09.002

MLA:

Ervedoza, Sylvain, and Enrique Zuazua. "Uniformly exponentially stable approximations for a class of damped systems." Journal De Mathematiques Pures Et Appliquees 91.1 (2009): 20-48.

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