Error analysis for Galerkin finite element approximations of general second order hyperbolic equations

Abstract

The vibration of elastic bodies and structures consisting of elastic bodies is an active research field in Applied Mathematics and Engineering. A mathematical model for such a vibrating system is a complex system of partial differential equations. However, the variational form is in each case of the same type as the variational form for the well known wave equation. The generalized problem is to solve a second order differential equation for a function with values in a Hilbert space. In this thesis this problem is referred to as the general second order hyperbolic equation. Obviously numerical approximation of solutions is inevitable. In this thesis we consider the continuous Galerkin finite element approximation of the solution. The aim in the research is to generalize the theory of convergence for hyperbolic problems to accommodate complex systems. A previous attempt to generalize the theory was made in a thesis in 2000, but the result is only valid for modal damping. This thesis and two articles that had a significant impact on the literature were considered for the present investigation. The methods in the two articles were successfully generalized and the results in the thesis slightly improved to create a comprehensive convergence theory. In 2002 an article on existence for the general second order hyperbolic equation appeared. The existence results are presented in variational form, convenient for the Galerkin method. Furthermore, different types of damping and their significance for the theory are identified. In this thesis the variational form of the general second order hyperbolic equation is used to create a general framework for convergence theory. It transpired that sufficient conditions for existence are also used to prove convergence. As for existence theory, the type of damping needed to be taken into account for convergence theory. This thesis is not merely about generalization, other contributions to the theory are also made. In all publications considered, error estimates are derived for the semi discrete approximation and then for the fully discrete approximation without using the results already obtained. We derive an error estimate for the semi-discrete approximation and then an estimate for the error in approximating the solution of the semi-discrete problem by the fully discrete approximation with respect to the same norm. The final estimate follows trivially by the triangle inequality. This approach has two advantages. It is not necessary to assume the existence of a third or fourth order derivative for the exact solution and the convergence analysis for the fully discrete approximation is simplified. It is well known that the smoothness of a solution is important in convergence analysis. In this thesis great care is taken to formulate regularity assumptions. Numerous applications are given in the thesis where it is clear that it is not trivial to apply the general results. This thesis also includes a number of new existence results.