Mesh adaptivity is a useful tool for efficient solution to partial differential equations in very complex geometries. In the present paper we discuss the use of polygonal mesh refinement in order to tackle two common issues: first, adaptively refine a provided good quality polygonal mesh preserving quality, second, improve the quality of a coarse poor quality polygonal mesh during the refinement process.

For finite element methods and triangular meshes, convergence of a posteriori mesh refinement algorithms and optimality properties have been widely investigated, whereas convergence and optimality are still open problems for polygonal adaptive methods.

In this article, we propose a new refinement method for convex cells with the aim of introducing some properties useful to tackle convergence and optimality for adaptive methods. The key issues in refining convex general polygons are: a refinement dependent only on the marked cells for refinement at each refinement step; a partial quality improvement, or, at least, a non degenerate quality of the mesh during the refinement iterations; a bound on the number of unknowns of the discrete problem with respect to the number of the cells in the mesh. Although these properties are quite common for refinement algorithms of triangular meshes, these issues are still open problems for polygonal meshes.

网格自适应性是有效求解非常复杂几何中的偏微分方程的有用工具。在本文中，我们讨论了多边形网格细化的使用，以解决两个常见问题：首先，自适应地细化提供的优质多边形网格，保持质量，其次，在细化过程中提高粗糙质量差的多边形网格的质量。

对于有限元方法和三角形网格，后验网格细化算法的收敛性和最优性已被广泛研究，而收敛性和最优性仍然是多边形自适应方法的未解决问题。

在本文中，我们提出了一种新的凸单元细化方法，旨在引入一些有助于解决自适应方法的收敛性和最优性的属性。细化凸一般多边形的关键问题是：在每个细化步骤中仅依赖于标记单元进行细化的细化；在细化迭代期间，部分质量改进，或者至少是网格的非退化质量；离散问题的未知数相对于网格中单元数的界限。尽管这些属性在三角网格的细化算法中很常见，但这些问题对于多边形网格仍然是悬而未决的问题。