The metric tensor of a Riemannian manifold can be approximated using Regge finite elements and such approximations can be used to compute approximations to the Gauss curvature and the Levi-Civita connection of the manifold. It is shown that certain Regge approximations yield curvature and connection approximations that converge at a higher rate than previously known. The analysis is based on covariant (distributional) curl and incompatibility operators which can be applied to piecewise smooth matrix fields whose tangential-tangential component is continuous across element interfaces. Using the properties of the canonical interpolant of the Regge space, we obtain superconvergence of approximations of these covariant operators. Numerical experiments further illustrate the results from the error analysis.

翻译：使用 Regge 有限元素可以近似Riemannian 元体的光度强。 这种近似值可用于计算高斯曲线的近似值和多元体的Levi-Civita 连接。 显示某些 Regge 近似值的曲线和连接近似值比以前已知的要高。 分析基于共变( 分布) 卷曲和不兼容操作器, 它可以应用于元素界面相近- 切异部分连续的平滑矩阵字段。 我们利用Regge 空间的共振间插特性, 获得了这些共变操作器近似值的超级一致。 数字实验进一步说明了错误分析的结果 。