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Boundary observability for the space semi-discretizations of the 1-D wave equation. (English) Zbl 0947.65101
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.

MSC:
65M20 Method of lines for initial value and initial-boundary value problems involving PDEs
35L05 Wave equation
93B07 Observability
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
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