Arch beam models: finite element analysis and superconvergence

Abstract

This paper studies finite element methods for a class of arch beam models. For both standard and mixed methods, existence and uniqueness results are proved, optimal rates of convergence are obtained and the superconvergence property is established. Reduced integration is shown to be an efficient method for arch beam problems and selected reduced integration is found to be identical to the mixed method. The significance of the analysis is threefold. The mixed method and the reduced integration methods converge uniformly at the optimal rate with respect to the arch thickness parameter, so they are locking free. Second, mixed method and reduced integration keep the superconvergence properties of the standard method. Finally, this is the first attempt to investigate the superconvergence of finite element methods for arch beam problems. We set up two types of superconvergence results: displacement at the nodal points and gradient at the Gauss points.