On recursive refinement of convex polygons

• Triangulation and quadrilateral refinements are well-known.
• A pentagonal refinement was introduced for solution of PDE by Floater and Lai.
• We show that one is not able to refine recursively a convex polygon with n  -side if n≥6n≥6.
• In addition, we introduce a refinement scheme to subdivide n  -gons by pentagons for n≥6n≥6.

It is known that one can improve the accuracy of the finite element solution of partial differential equations (PDE) by uniformly refining a triangulation. Similarly, one can uniformly refine a quadrangulation. Recently polygonal meshes have been used for numerical solution of partial differential equations based on virtual element methods, weak Galerkin methods, and polygonal spline methods. A refinement scheme of pentagonal partition was introduced in Floater and Lai (2016). It is natural to ask if one can create a hexagonal refinement or general polygonal refinement schemes. In this short article, we show that one cannot refine a convex hexagon using convex hexagons of smaller size. In general, we show that one can only refine a convex n-gon by convex n  -gons of smaller size if n≤5n≤5.