LEADER 05003nam 22008295 450 001 9910299977103321 005 20200629180017.0 010 $a1-4614-5811-0 024 7 $a10.1007/978-1-4614-5811-1 035 $a(CKB)3710000000093418 035 $a(SSID)ssj0001187553 035 $a(PQKBManifestationID)11673376 035 $a(PQKBTitleCode)TC0001187553 035 $a(PQKBWorkID)11258055 035 $a(PQKB)11276801 035 $a(MiAaPQ)EBC1698429 035 $a(DE-He213)978-1-4614-5811-1 035 $a(PPN)17782302X 035 $a(EXLCZ)993710000000093418 100 $a20140310d2014 u| 0 101 0 $aeng 135 $aurcnu|||||||| 181 $ctxt 182 $cc 183 $acr 200 10$aSymmetric Discontinuous Galerkin Methods for 1-D Waves $eFourier Analysis, Propagation, Observability and Applications /$fby Aurora Marica, Enrique Zuazua 205 $a1st ed. 2014. 210 1$aNew York, NY :$cSpringer New York :$cImprint: Springer,$d2014. 215 $a1 online resource (114 pages) $cillustrations (some color) 225 1 $aSpringerBriefs in Mathematics,$x2191-8198 300 $aBibliographic Level Mode of Issuance: Monograph 311 $a1-4614-5810-2 320 $aIncludes bibliographical references. 327 $a1. 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. 330 $aThis work describes the propagation properties of the so-called symmetric interior penalty discontinuous Galerkin (SIPG) approximations of the 1-d wave equation. This is done by means of linear approximations on uniform meshes. First, a careful Fourier analysis is constructed, highlighting the coexistence of two Fourier spectral branches or spectral diagrams (physical and spurious) related to the two components of the numerical solution (averages and jumps). Efficient filtering mechanisms are also developed by means of techniques previously proved to be appropriate for classical schemes like finite differences or P1-classical finite elements. In particular, the work presents a proof that the uniform observability property is recovered uniformly by considering initial data with null jumps and averages given by a bi-grid filtering algorithm. Finally, the book explains how these results can be extended to other more sophisticated conforming and non-conforming finite element methods, in particular to quadratic finite elements, local discontinuous Galerkin methods and a version of the SIPG method adding penalization on the normal derivatives of the numerical solution at the grid points.  This work is the first publication to contain a rigorous analysis of the discontinuous Galerkin methods for wave control problems. It will be of interest to a range of researchers specializing in wave approximations. 410 0$aSpringerBriefs in Mathematics,$x2191-8198 606 $aNumerical analysis 606 $aFourier analysis 606 $aApproximation theory 606 $aPartial differential equations 606 $aAlgorithms 606 $aApplied mathematics 606 $aEngineering mathematics 606 $aNumerical Analysis$3https://scigraph.springernature.com/ontologies/product-market-codes/M14050 606 $aFourier Analysis$3https://scigraph.springernature.com/ontologies/product-market-codes/M12058 606 $aApproximations and Expansions$3https://scigraph.springernature.com/ontologies/product-market-codes/M12023 606 $aPartial Differential Equations$3https://scigraph.springernature.com/ontologies/product-market-codes/M12155 606 $aAlgorithms$3https://scigraph.springernature.com/ontologies/product-market-codes/M14018 606 $aApplications of Mathematics$3https://scigraph.springernature.com/ontologies/product-market-codes/M13003 615 0$aNumerical analysis. 615 0$aFourier analysis. 615 0$aApproximation theory. 615 0$aPartial differential equations. 615 0$aAlgorithms. 615 0$aApplied mathematics. 615 0$aEngineering mathematics. 615 14$aNumerical Analysis. 615 24$aFourier Analysis. 615 24$aApproximations and Expansions. 615 24$aPartial Differential Equations. 615 24$aAlgorithms. 615 24$aApplications of Mathematics. 676 $a530.124 700 $aMarica$b Aurora$4aut$4http://id.loc.gov/vocabulary/relators/aut$0721756 702 $aZuazua$b Enrique$4aut$4http://id.loc.gov/vocabulary/relators/aut 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a9910299977103321 996 $aSymmetric Discontinuous Galerkin Methods for 1-D Waves$92512150 997 $aUNINA