This paper presents a higher-order approximation of polygonal elements using the generalized finite element method. The approximation is constructed using a virtual-node polygonal element based on the partition of unity coupled with the polynomial nodal approximation. Because the approximation functions are polynomials, the numerical integration can be evaluated accurately using Gauss quadrature. The proposed method passes the higher-order patch test and yields an optimal convergence rate for polygonal meshes. This higher-order method is also applied to the h-adaptive method on triangular quadtree mesh, which allows arbitrary-level hanging nodes. The study results demonstrate that the process of the h-adaptive refinement using the proposed method can deliver accuracy comparable with that of the red-green refinement while being even simpler, and this is demonstrated using several numerical tests for the Poisson problem.

polygonal element, generalized finite element method, virtual node method, triangular quadtree mesh, hanging nodes.