High-order in time discontinuous Galerkin finite element methods for linear wave equations

Abstract

In this thesis we analyse the high-order in time discontinuous Galerkin nite element method (DGFEM) for second-order in time linear abstract wave equations. Our abstract approximation analysis is a generalisation of the approach introduced by Claes Johnson (in Comput. Methods Appl. Mech. Engrg., 107:117-129, 1993), writing the second order problem as a system of fi rst order problems. We consider abstract spatial (time independent) operators, highorder in time basis functions when discretising in time; we also prove approximation results in case of linear constraints, e.g. non-homogeneous boundary data. We take the two steps approximation approach i.e. using high-order in time DGFEM; the discretisation approach in time introduced by D Schötzau (PhD thesis, Swiss Federal institute of technology, Zürich, 1999) to fi rst obtain the semidiscrete scheme and then conformal spatial discretisation to obtain the fully-discrete formulation. We have shown solvability, unconditional stability and conditional a priori error estimates within our abstract framework for the fully discretized problem. The skew-symmetric spatial forms arising in our abstract framework for the semi- and fully-discrete schemes do not full ll the underlying assumptions in D. Schötzau's work. But the semi-discrete and fully discrete forms satisfy an Inf-sup condition, essential for our proofs; in this sense our approach is also a generalisation of D. Schötzau's work. All estimates are given in a norm in space and time which is weaker than the Hilbert norm belonging to our abstract function spaces, a typical complication in evolution problems. To the best of the author's knowledge, with the approximation approach we used, these stability and a priori error estimates with their abstract structure have not been shown before for the abstract variational formulation used in this thesis. Finally we apply our abstract framework to the acoustic and an elasto-dynamic linear equations with non-homogeneous Dirichlet boundary data.