Although Regge finite element functions are not continuous, useful generalizations of nonlinear derivatives like the curvature, can be defined using them. This paper is devoted to studying the convergence of the finite element lifting of a generalized (distributional) Gauss curvature defined using a metric tensor in the Regge finite element space. Specifically, we investigate the interplay between the polynomial degree of the curvature lifting by Lagrange elements and the degree of the metric tensor in the Regge finite element space. Previously, a superconvergence result, where convergence rate of one order higher than expected, was obtained when the metric is the canonical Regge interpolant of the exact metric. In this work, we show that an even higher order can be obtained if the degree of the curvature lifting is reduced by one polynomial degre and if at least linear Regge elements are used. These improved convergence rates are confirmed by numerical examples.

翻译：尽管Regge有限元函数不连续，但可利用它们定义曲率等非线性导数的有用推广。本文致力于研究利用Regge有限元空间的度量张量定义的广义（分布）高斯曲率的有限元提升收敛性。具体而言，我们探讨了拉格朗日单元的曲率提升多项式次数与Regge有限元空间度量张量次数之间的相互作用。此前，当度量为精确度量的标准Regge插值时，获得了预期高一阶收敛的超收敛结果。本研究表明，若将曲率提升的次数降低一个多项式阶，且至少使用线性Regge单元，则可获得更高阶的收敛性。数值算例验证了这些改进的收敛速率。