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Numerical meshes ensuring uniform observability of one-dimensional waves: construction and analysis. (English) Zbl 1433.65157

Summary: We build nonuniform numerical meshes for the finite difference and finite element approximations of the one-dimensional wave equation, ensuring that all numerical solutions reach the boundary, as continuous solutions do, in the sense that the full discrete energy can be observed by means of boundary measurements, uniformly with respect to the mesh size. The construction of the nonuniform mesh is achieved by means of a concave diffeomorphic transformation of a uniform grid into a nonuniform one, making the mesh finer and finer when approaching the right boundary. For uniform meshes it is known that high-frequency numerical wave packets propagate very slowly without ever getting to the boundary. Our results show that this pathology can be avoided by taking suitable nonuniform meshes. This also allows us to build convergent numerical algorithms for the approximation of boundary controls of the wave equation.

MSC:

65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
49M25 Discrete approximations in optimal control
65M50 Mesh generation, refinement, and adaptive methods for the numerical solution of 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
93B07 Observability
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