Anisotropic mesh quality measures and adaptation are studied for convex polygonal meshes. Three sets of alignment and equidistribution measures are developed, based on least squares fitting, generalized barycentric mappings, and the singular value decomposition, respectively. Numerical tests show that all three sets of mesh quality measures provide good measurements for the quality of polygonal meshes under given anisotropic metrics. Based on the second set of quality measures and using a moving mesh partial differential equation, an anisotropic adaptive polygonal mesh method is constructed for the numerical solution of second-order elliptic equations. Numerical examples are presented to demonstrate the effectiveness of the method.

研究了凸多边形网格的各向异性网格质量度量和适应性。分别基于最小二乘拟合，广义重心映射和奇异值分解，开发了三组对齐和均分度量。数值测试表明，在给定的各向异性指标下，所有三组网格质量度量都可以很好地度量多边形网格的质量。基于第二组质量测度，并使用移动网格偏微分方程，构造了各向异性自适应多边形网格方法来求解二阶椭圆方程的数值解。数值算例表明了该方法的有效性。