On recursive refinement of convex polygons

It is known that one can improve the accuracy of the finite element solution of partial differential equation (PDE) by uniformly refining a triangulation. Similarly, one can uniformly refine a quadrangulation. Recently a refinement scheme of pentagonal partition was introduced in [4]. It is demonstrated that the numerical solution of Poisson equation based on the pentagonal refinement scheme outperforms the solutions based on the traditional triangulation refinement scheme as well as quadrangulation refinement scheme. It is natural to ask if one can create a hexagonal refinement or general polygonal refinement schemes with a hope to offer even further improvement. In this short article, we show that one cannot refine a hexagon using hexagons of smaller size. In general, one can only refine an n-gon by n-gons of smaller size if n ≤ 5. Furthermore, we introduce a refinement scheme of a general polygon based on the pentagon scheme.