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Zbl 0947.65101
Infante, Juan Antonio; Zuazua, Enrique
Boundary observability for the space semi-discretizations of the 1-D wave equation. (English)
[J] M2AN, Math. Model. Numer. Anal. 33, No.2, 407-438 (1999). ISSN 0764-583X; ISSN 1290-3841/e

The numerical solution of the one-dimensional wave equation with homogeneous Dirichlet boundary conditions is considered especially by both space semi-discretizations: finite difference and finite element methods. The problem of boundary observability, i.e., the problem of wheather the total energy of solutions can be estimated uniformly in terms of the energy concentrated on the boundary as the mesh discretization $h \rightarrow 0$ is investigated. Due to the spurios modes that the numerical scheme introduces at high frequencies, there is no such a uniform bound. A uniform bound is proved in a subspace of solutions generated by the low frequencies of the discrete system. When $h \rightarrow 0$ these finite-dimensional spaces increase and eventually cover the whole space. Thus the well-known observability property of the continuous system as the limit of discrete observability estimates as the mesh size tends to zero is recovered.
[Angela Handlovičová (Bratislava)]

MSC 2000:: *65M20 Method of lines (IVP of PDE)
35L05 Wave equation
93B07 Observability
65M06 Finite difference methods (IVP of PDE)
65M60 Finite numerical methods (IVP of PDE)

Keywords: wave equation; semi-discretization; finite difference; finite element; boundary observability

Cited in: Zbl 1176.93017 Zbl 1007.93037

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