Finite element approximation of general second order hyperbolic equations

Abstract

The vibration of elastic bodies and structures consisting of elastic bodies is an active research field in engineering. A mathematical model for such a vibrating system is a complex system of partial differential equations. In this dissertation only linear cases are considered and the variational form is of the same type as the variational form for the well known wave equation in each case. The generalized problem is to solve a second order differential equation for a function with values in a Hilbert space. This problem is referred to as the general second order hyperbolic equation in this dissertation. Obviously numerical approximation of solutions is inevitable. The finite element method proved to be ideal. It is used for steady-state problems, eigenvalue problems and dynamic problems. This dissertation is a study of the state of the theory for dynamic problems: convergence of the finite element approximation and error estimates. The approach is to split the approximation problem. First the convergence for the semi-discrete problem is considered followed by the convergence of the fully discrete approximation to the solution of the semi-discrete problem. The focus is on the (continuous) Galerkin method. However one chapter is devoted to the application of the Trotter-Kato theorem and brief discussions on the mixed finite element method and discontinuous Galerkin method are included. An in depth study was made of two articles and the relevant part of a PhD thesis. From a further five articles the useful parts (for this dissertation) were studied in depth. Five more articles were read but only to determine their contribution to the theory. In this dissertation the theory for the Galerkin method is presented as a unit; the material from the sources are merged, duplication eliminated and similarities and differences noted. By standardizing the notation and expanding proofs the presentation is (hopefully) more readable than the original publications. Other notable contributions in this dissertation are the generalization of certain assumptions for wider application, weakening of some assumptions, the generalization of the main result in the 1976 article of Baker, application of Dupont's method (1973) to the Timoshenko beam with boundary damping and application of the Trotter-Kato theorem to the general second order hyperbolic equation. The main conclusion is that the general results are not satisfactory. Apart from the application of the Trotter-Kato theorem, where error estimates are in general not available, the theory is not sufficiently general to cover all linear vibration models. It is also clear from examples that the assumptions in the theory are too restrictive. The dissertation does not only provide an overview of the theory of the Galerkin method for dynamic problems but the presentation provides a basis for future research.