On the Quadratic Finite Element Approximation of 1D Waves: Propagation, Observation, Control, and Numerical Implementati
In arXiv:1112.4297 , we studied the propagation, observation, and control properties of the 1D wave equation on a bounded interval semi-discretized in space using the quadratic classical finite element approximation. It was shown that the discrete wave dy
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Abstract In arXiv:1112.4297, we studied the propagation, observation, and control properties of the 1D wave equation on a bounded interval semi-discretized in space using the quadratic classical finite element approximation. It was shown that the discrete wave dynamics consisting of the interaction of nodal and midpoint components leads to the existence of two different eigenvalue branches in the spectrum: an acoustic one, of physical nature, and an optic one, of spurious nature. The fact that both dispersion relations have critical points where the corresponding group velocities vanish produces numerical wave packets whose energy is concentrated in the interior of the domain, without propagating, and for which the observability constant blows up as the mesh size goes to zero. This extends to the quadratic finite element setting the fact that the classical property of continuous waves being observable from the boundary fails for the most classical approximations on uniform meshes (finite differences, linear finite elements, etc.). As a consequence, the numerical controls of minimal norm may blow up as the mesh size parameter tends to zero. To cure these high-frequency pathologies, in arXiv:1112.4297 we designed a filtering mechanism consisting in taking piecewise linear and continuous initial data (so that the curvature component vanishes at the initial time) with nodal components given by a bi-grid algorithm. The aim of this article is to implement this filtering technique and to show numerically its efficiency. Keywords Linear and quadratic finite element method · Uniform mesh · Vanishing group velocity · Observability/controllability property · Acoustic/optic mode · Bi-grid algorithm · Conjugate gradient algorithm A. Marica · E. Zuazua BCAM—Basque Center for Applied Mathematics, Alameda de Mazarredo 14, 48009, Bilbao, Basque Country, Spain A. Marica e-mail: [email protected] E. Zuazua () Ikerbasque—Basque Foundation for Science, Alameda Urquijo 36-5, Plaza Bizkaia, 48011, Bilbao, Basque Country, Spain e-mail: [email protected] C.A. de Moura, C.S. Kubrusly (eds.), The Courant–Friedrichs–Lewy (CFL) Condition, DOI 10.1007/978-0-8176-8394-8_6, © Springer Science+Business Media New York 2013
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1 Preliminaries on the Continuous Model and Problem Formulation Consider the 1D wave equation with non-homogeneous boundary conditions: ⎧ ⎪ ⎨ytt (x, t) − yxx (x, t) = 0, x ∈ (0, 1), t > 0, y(0, t) = 0, y(1, t) = v(t), t > 0, ⎪ ⎩ yt (x, 0) = y 1 (x), x ∈ (0, 1). y(x, 0) = y 0 (x),
(1)
System (1) is said to be exactly controllable in time T ≥ 2 if, for all (y 0 , y 1 ) ∈ L2 × H −1 (0, 1), there exists a control function v ∈ L2 (0, T ) such that the solution of (1) can be driven to rest at time T , i.e. y(x, T ) = yt (x, T ) = 0. We also introduce the adjoint 1D wave equation with homogeneous boundary conditions: ⎧ ⎪ ⎨utt (x, t) − uxx (x, t) = 0, x ∈ (0, 1), t > 0, (2) u(0, t) = u(1, t) = 0, t > 0, ⎪ ⎩ ut (x, T ) = u1 (x), x ∈ (0, 1). u(x, T ) = u0 (x), This system is well known to be well pos
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