We construct and analyze piecewise approximations of functional data on arbitrary 2D bounded domains using generalized barycentric finite elements, and particularly quadratic serendipity elements for planar polygons. We compare approximation qualities (precision/convergence) of these partition-of-unity finite elements through numerical experiments, using Wachspress coordinates, natural neighbor coordinates, Poisson coordinates, mean value coordinates, and quadratic serendipity bases over polygonal meshes on the domain. For a convex n-sided polygon, the quadratic serendipity elements have 2n basis functions, associated in a Lagrange-like fashion to each vertex and each edge midpoint, rather than the usual $n(n+1)/2$ basis functions to achieve quadratic convergence. Two greedy algorithms are proposed to generate Voronoi meshes for adaptive functional/scattered data approximations. Experimental results show space/accuracy advantages for these quadratic serendipity finite elements on polygonal domains versus traditional finite elements over simplicial meshes. Polygonal meshes and parameter coefficients of the quadratic serendipity finite elements obtained by our greedy algorithms can be further refined using an $L_2$-optimization to improve the piecewise functional approximation. We conduct several experiments to demonstrate the efficacy of our algorithm for modeling features/discontinuities in functional data/image approximation.

我们使用广义重心有限元，尤其是平面多边形的二次偶然性元素，在任意2D有界域上构造和分析函数数据的分段逼近。我们通过数值实验，使用Wachspress坐标，自然邻域坐标，泊松坐标，平均值坐标和域上多边形网格的二次偶然性基数，通过数值实验比较这些整体划分有限元的近似质量（精度/收敛）。对于凸的n面多边形，二次奇异元素具有2 n个基函数，以类似于拉格朗日的方式与每个顶点和每个边中点相关联，而不是通常的$ñ（ñ+\mathrm{1个}）/2$基本函数以实现二次收敛。提出了两种贪婪算法来生成Voronoi网格，用于自适应功能/分散数据逼近。实验结果表明，与单纯网格相比，多边形域上的这些二次偶然性有限元相对于传统有限元在空间/准确性上具有优势。通过我们的贪婪算法获得的二次偶然性有限元的多边形网格和参数系数可以使用${\mathrm{大号}}_2$-优化以改进分段函数逼近。我们进行了一些实验，以证明我们的算法在功能数据/图像逼近中对特征/不连续性建模的有效性。