We construct finite element approximations of the Levi-Civita connection and its curvature on triangulations of oriented two-dimensional manifolds. Our construction relies on the Regge finite elements, which are piecewise polynomial symmetric (0, 2)-tensor fields possessing single-valued tangential-tangential components along element interfaces. When used to discretize the Riemannian metric tensor, these piecewise polynomial tensor fields do not possess enough regularity to define connections and curvature in the classical sense, but we show how to make sense of these quantities in a distributional sense. We then show that these distributional quantities converge in certain dual Sobolev norms to their smooth counterparts under refinement of the triangulation. We also discuss projections of the distributional curvature and distributional connection onto piecewise polynomial finite element spaces. We show that the relevant projection operators commute with certain linearized differential operators, yielding a commutative diagram of differential complexes.

我们构建了 Levi-Civita 连接的有限元近似及其在定向二维流形三角剖分上的曲率。我们的构造依赖于 Regge 有限元，它是分段多项式对称 (0, 2)-张量场，沿元素界面具有单值切向-切向分量。当用于离散化黎曼度量张量时，这些分段多项式张量场不具备足够的规律性来定义经典意义上的连接和曲率，但我们展示了如何在分布意义上理解这些量。然后，我们证明这些分布量在某些对偶 Sobolev 范数中收敛到它们在三角剖分细化下的平滑对应物。我们还讨论了分布曲率和分布连接在分段多项式有限元空间上的投影。我们表明相关投影算子与某些线性化微分算子交换，产生微分复形的交换图。