We perform a complete Fourier analysis of the semi-discrete 1-d wave equationobtained through a P1 discontinuous Galerkin (DG) approximation of thecontinuous wave equation on an uniform grid. The resulting system exhibits theinteraction of two types of components: a physical one and a spurious one,related to the possible discontinuities that the numerical solution allows.Each dispersion relation contains critical points where the corresponding groupvelocity vanishes. Following previous constructions, we rigorously build wavepackets with arbitrarily small velocity of propagation concentrated either onthe physical or on the spurious component. We also develop filtering mechanismsaimed at recovering the uniform velocity of propagation of the continuoussolutions. Finally, some applications to numerical approximation issues ofcontrol problems are also presented.
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