Convergence analysis of finite element approximations of solutions of hyperbolic equations

Abstract

The vibration of elastic bodies and structures consisting of elastic bodies is an active research field in engineering and applied mathematics. Typically, a mathematical model is a complex system of partial differential equations. However, a model problem may not have a solution in the classical sense and the rate of convergence of numerical approximations depends on the “smoothness” of a solution. The aim of this research is to investigate the disparity noticed in the theory between the existence of solutions and the regularity assumed on these solutions for convergence of the Galerkin finite element method. In the articles considered, substantially more differentiability properties for the solution are assumed than obtained in existence theory. These assumptions are very restrictive; the solution is required to be smoother than even a classical solution. The theory of existence of a solution to a general linear vibration problem that appeared in an article published in 2002, was considered first. To compare, alternative theories on existence of solutions to hyperbolic partial differential equations, were also studied. The existence results, improved regularity of solutions and compatibility conditions, which are highly restrictive, are presented. In 2013 an article appeared wherein convergence is proved, but with weaker assumptions than the other articles considered. This is achieved by splitting the error into the semi-discreet and fully discreet errors. However, it is still necessary to assume higher regularity of the solution. The focus in this dissertation was to compare the article to other research results, and to highlight significant parts of the proofs in the article. Also, minor improvements were made and it was proved that the results obtained from existence theory are sufficient for convergence, but no result on the order of convergence could be obtained. A recent article (2011) on the continuous Galerkin method, where the model problem considered includes strong damping, was also analysed. The results from this article is proved in great detail, and possible oversights or omissions discovered are either rectified or reported. The discontinuous Galerkin (DG) finite element method is also included in the research, with the aim to determine whether the assumptions made on the regularity needed for convergence are less restrictive than those made for the continuous Galerkin method. Disappointingly, the results offer no significant improvement. The semi-discreet and fully discreet DG error estimates are from articles published in 2006 and 2009 respectively. The results are proven in greater detail in this dissertation. Interesting phenomena obtained from numerical experiments are observed and to a large degree the theory and experiments agree. However, there are indications that the order of convergence may in some cases be better than predicted by the theory. The main conclusion is that there is a problem when applying theoretical results to real world problems. Further research is required to prove results where error estimates are derived without restrictive assumptions on boundary and initial data.