In this work we present polygonal finite element method (Poly-FEM) for the analysis of two dimensional plane elasticity problems. The generation of meshes consisting of n− sided polygonal finite elements is based on the generation of a centroidal Voronoi tessellation (CVT). An unstructured tessellation of a scattered point set, that minimally covers the proximal space around each point in the point set is generated whereby the method also includes tessellation of nonconvex domains.In this work, a patch recovery type of stress smoothing technique that utilizes polygonal element patches for obtaining smooth stresses is proposed for obtaining the smoothed finite element stresses. A recovery type a − posteriori error estimator that estimates the energy norm of the error from the recovered solution is then adopted for the polygonal finite element method. The refinement of the polygonal elements is then made on an region by region basis through a refinement index. For the numerical integration of the Galerkin weak form over polygonal finite element domains we resort to classical Gaussian quadrature applied to triangular sub domains of each polygonal element
