Overview

This workdescribes the propagation properties of the so-called symmetric interior penalty discontinuous Galerkin (SIPG) approximations of the 1-d wave equation. This is done by means of linear approximations on uniform meshes. First, a careful Fourier analysis is constructed, highlightingthe coexistence of two Fourier spectral branches or spectral diagrams (physical and spurious) related to the two components of the numerical solution (averages and jumps). Efficient filtering mechanisms are also developedby means of techniques previously proved to be appropriate for classical schemes like finite differences or P1-classical finite elements. In particular, the work presents a proof thatthe uniform observability property is recovered uniformly by considering initial data with null jumps and averages given by a bi-grid filtering algorithm. Finally, the bookexplains how theseresults can be extended to other more sophisticated conforming and non-conforming finite element methods, in particular to quadratic finiteelements, local discontinuous Galerkin methods and a version of the SIPG method adding penalization on the normal derivatives of the numerical solution at the grid points. This work is the first publication tocontaina rigorous analysis of the discontinuous Galerkin methods for wave control problems. It will be of interest to a range of researchers specializing inwave approximations.

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9781306692618

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