Title Symmetric discontinuous Galerkin methods of 1-D waves : Fourier analysis, propagation, observability and applications / by Aurora Marica, Enrique Zuazua

Published New York : Springer, 2014

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Description 1 online resource Series SpringerBriefs in mathematics SpringerBriefs in mathematics Contents 1. Preliminaries -- 2. Discontinuous Galerkin approximations and main results -- 3. Bibliographical notes -- 4. Fourier analysis of the DG methods -- 5. Non-uniform observability for DG approximations of waves -- 6. Filtering mechanisms -- 7. Extensions to other numerical approximation schemes -- 8. Comments and open problems -- A technical proof -- References Summary 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 Analysis wiskunde mathematics partial differential equations numerieke methoden numerical methods approximaties approximations fourieranalyse fourier analysis algoritmen algorithms toegepaste wiskunde applied mathematics Mathematics (General) Wiskunde (algemeen) Notes Print version record Subject Waves -- Mathematics Galerkin methods. Approximation theory. SCIENCE -- Energy. SCIENCE -- Mechanics -- General. SCIENCE -- Physics -- General. Approximation theory Galerkin methods Waves -- Mathematics Form Electronic book Author Zuazua, E. (Enrique), author ISBN 9781461458111 1461458110

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