Tesi di LAUREA SPECIALISTICA Titolo An adaptive discontinuous Galerkin spectral element methods for systems of ordinary differential equations with applications to elastodynamics Data 2014-12-19 Autore/i Dal Santo, Niccolo Relatore Quarteroni, A. Relatore Antonietti, P.F. Correlatore Mazzieri, I. Full text non disponibile Abstract The aim of this Master Thesis is to propose and analyze a new high order finite element method for the time integration of second order systems of ordinary differential equations. This kind of equations typically arise after the space semi-discretization of hyperbolic problems, e.g., wave equation, elastodynamics, acoustics. In this work, we develop a high order scheme based on the discontinuous Galerkin spectral element method (DGSE) for the time integration of such kind of systems. This approach allows us to combine the flexibility of the discontinuous Galerkin method with the high order accuracy of the spectral element technique. After introducing the method, we analyze its well-posedness and present an ext{a-priori} error estimate. Then, we assess its performance on both simplified test cases as well as on a problem of practical interest, namely to integrate the (second order) system of equations arising from the semi-discretization of the elastodynamics equation with a DGSE method. Our numerical computations have been obtained with the computational code SPEED (http://mox.polimi.it/it/progetti/speed/). Finally, we develop an adaptive technique to tune the local polynomial degree of the solution in every time interval and test this algorithm on some meaningful examples.