Error estimation for the polygonal finite element method for smooth and singular linear elasticity

A recovery-based error indicator developed to evaluate the quality of polygonal finite element approximations is presented in this paper. Generalisations of the finite element method to arbitrary polygonal meshes have been increasingly investigated in the last years, as they provide flexibility in meshing and improve solution accuracy. As any numerical approximation, they have an induced error which has to be accounted for in order to validate the approximate solution. Here, we propose a recovery type error measure based on a moving least squares fitting of the finite element stress field. The quality of the recovered field is improved by imposing equilibrium conditions and, for singular problems, splitting the stress field into smooth and singular parts. We assess the performance of the error indicator using three problems with exact solution, and we also compared the results with those obtained with standard finite element meshes based on simplexes. The results indicate good values for the local and global effectivities, similar to the values obtained for standard approximations, and are always within the recommended range.