Numerical approximation of the boundary control for the wave equation with mixed finite elements in a square

By C. Castro, Sorin Daniel Micu, Arnaud Münch. 2007

47 citations recorded by the source corpus.

This paper studies the numerical approximation of the boundary control for the wave equation in a square domain. It is known that the discrete and semi-discrete models ob- tained by discretizing the wave equation with the usual ¯nite di®erence or ¯nite element methods do not provide convergent sequences of approximations to the boundary control of the continuous wave equation, as the mesh size goes to zero (see [7, 15]). Here we introduce and analyze a new semi-discrete model based on the space discretization of the wave equa- tion using a mixed ¯nite element method with two di®erent basis functions for the position and velocity. The main theoretical result is a uniform observability inequality which allows us to construct a sequence of approximations converging to the minimal L2inorm control of the continuous wave equation. We also introduce a fully-discrete system, obtained from our semi-discrete scheme, for which we conjecture that it provides a convergent sequence of discrete approximations as both h and ¢t, the time discretization parameter, go to zero. We illustrate this fact with several numerical experiments.

Subfields: Computational Mechanics, Computational Theory and Mathematics, Control and Systems Engineering

View Paper: https://www.semanticscholar.org/paper/9d55190fdbe999871ceefb1e8b97d659a69faaf6

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Computational Mechanics

1. Sylvain Ervedoza
2. Enrique Zuazua
3. Nazim Idrisoglu Mahmudov
4. Emmanuel Trélat
5. Yacine Chitour

In Computational Theory and Mathematics

1. Sylvain Ervedoza
2. Enrique Zuazua
3. Xu Zhang
4. Marius Tucsnak
5. Genni Fragnelli
6. Luc Miller
7. Piermarco Cannarsa
8. Nazim Idrisoglu Mahmudov

In Control and Systems Engineering

1. Sylvain Ervedoza
2. Enrique Zuazua
3. Xu Zhang
4. Karine Beauchard
5. Luc Miller
6. Marius Tucsnak
7. Genni Fragnelli
8. Piermarco Cannarsa

Citations in the corpus

• Uniformly exponentially stable approximations for a class of damped systems (2009, methodology): “1) as [5,6,13,11, 24,27,23] (see [32] for more references).”
• The Wave Equation: Control and Numerics (2012, methodology): “This is the case for instance for the mixed finite element method [41, 17, 18, 28].”