We introduce a finite element construction for use on the class of convex, planar polygons and show it obtains a quadratic error convergence estimate. On a convex n-gon, our construction produces 2n basis functions, associated in a Lagrange-like fashion to each vertex and each edge midpoint, by transforming and combining a set of n(n + 1)/2 basis functions known to obtain quadratic convergence. The technique broadens the scope of the so-called 'serendipity' elements, previously studied only for quadrilateral and regular hexahedral meshes, by employing the theory of generalized barycentric coordinates. Uniform a priori error estimates are established over the class of convex quadrilaterals with bounded aspect ratio as well as over the class of convex planar polygons satisfying additional shape regularity conditions to exclude large interior angles and short edges. Numerical evidence is provided on a trapezoidal quadrilateral mesh, previously not amenable to serendipity constructions, and applications to adaptive meshing are discussed.

中文翻译：

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使用广义重心坐标的多边形上的二次偶然性有限元

我们介绍了一种用于凸面平面多边形类的有限元构造，并表明它获得了二次误差收敛估计。在凸 n 边形上，我们的构造产生 2n 个基函数，通过转换和组合一组已知的 n(n + 1)/2 基函数来获得二次方，以类似拉格朗日的方式关联到每个顶点和每个边中点收敛。该技术通过采用广义重心坐标理论扩大了所谓的“意外”元素的范围，以前仅针对四边形和规则六面体网格进行了研究。在具有有界纵横比的凸四边形类以及满足附加形状规则性条件以排除大内角和短边的凸平面多边形类上建立统一先验误差估计。在梯形四边形网格上提供了数值证据，以前不适用于意外构造，并讨论了自适应网格划分的应用。