On the Lack of Uniform Observability for Discontinuous Galerkin Approximations of Waves

The purpose of this chapter is to construct initial data for the discontinuous Galerkin approximations of the wave equation so that the corresponding wave packets propagate arbitrarily slowly. Our construction is based on the fact that on each Fourier mode there are critical wave numbers at which the corresponding group velocity vanishes. Moreover, we prove that the observability constant blows up at least polynomially at any order as the mesh size h tends to zero.

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