On the Quadratic Finite Element Approximation of 1D Waves: Propagation, Observation, Control, and Numerical Implementation

Marica, Aurora; Zuazua, Enrique

doi:10.1007/978-0-8176-8394-8_6

Cited by 9 publications

(3 citation statements)

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“…Multiplying system (17) by any solution U h (t) of the adjoint problem (10), integrating in time and imposing that at t = T the solution is at rest, i.e. (18) (Y h,t (T ),…”

Section: The Overline Symbol Denotes Complex Conjugation)mentioning

confidence: 99%

“…Moreover, the Euler-Lagrange equation ( 21) associated to J h characterizes the optimal control (22). Remark that when the space of initial data in ( 10) is restricted to a subspace S h ⊂ V h , for example the ones given by (55) or (56), the exact controllability condition (18) holds for all (U 0 h , U 1 h ) ∈ S h . This does not imply that the final state (Y h,t (T ), −Y h (T )) in the controlled problem (17) is exactly controllable to the rest, but its orthogonal projection Γ S h from V h on the subspace S h , i.e.…”

Section: Convergence Of the Discrete Controlsmentioning

confidence: 99%

“…This is due to the fact that |λ h (ξ) − ξ| ∼ h 2 ξ 3 , where λ h (ξ) can be each one of the dispersion relations for the finite difference or finite element approximation of the wave equation. In [18], we observed the fact that the acoustic dispersion relation λ a h (ξ) of the P 2 -finite element method approximates the continuous one ξ with error order h 4 ξ 5 for all ξ ∈ [0, π/h], so that the convergence error for the numerical controls obtained by the bi-grid algorithm in the quadratic approximation of the wave equation increases to h 4/5 under the same regularity assumptions on the continuous initial data to be controlled.…”

Section: (93)mentioning

confidence: 99%

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On the Quadratic Finite Element Approximation of One-Dimensional Waves: Propagation, Observation, and Control

Marica¹,

Zuazua²

2012

SIAM J. Numer. Anal.

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We study the propagation, observation and control properties of the 1 − d wave equation on a bounded interval discretized in space using the quadratic P 2 -classical finite element approximation. A careful Fourier analysis of the discrete wave dynamics reveals two different branches in the spectrum: the acoustic one, of physical nature, and the optic one, related to the perturbations that this second-order finite element approximation introduces with respect to the P 1 one. On both modes there are high frequencies with vanishing group velocity as the mesh size tends to zero. This shows that the classical property of continuous waves of being observable from the boundary fails to be uniform for this discretization scheme. As a consequence of this, the controls of the discrete waves may blow-up as the mesh size tends to zero. To remedy these high frequency pathologies, we design filtering mechanisms based on a bi-grid algorithm for which one can recover the uniformity of the observability constant in a finite time and, consequently, the possibility to control with uniformly bounded L 2 -controls appropriate projections of the solutions. This also allows showing that, by relaxing the control requirement, the controls are uniformly bounded and converge to the continuous ones as the mesh size tends to zero.

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“…Multiplying system (17) by any solution U h (t) of the adjoint problem (10), integrating in time and imposing that at t = T the solution is at rest, i.e. (18) (Y h,t (T ),…”

Section: The Overline Symbol Denotes Complex Conjugation)mentioning

confidence: 99%

Section: Convergence Of the Discrete Controlsmentioning

confidence: 99%

Section: (93)mentioning

confidence: 99%

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