Finite Element Approximation of the Levi-Civita Connection and Its Curvature in Two Dimensions
Finite Element Approximation of the Levi-Civita Connection and Its Curvature in Two Dimensions
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DOI：
10.1007/s10208-022-09597-1
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发表时间：
2021-11
影响因子：
3
通讯作者：
Yakov Berchenko-Kogan;Evan S. Gawlik
中科院分区：
文献类型：
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作者：
Yakov Berchenko-Kogan;Evan S. Gawlik
我们构造了Levi-Civita联络及其曲率在定向二维流形三角剖分上的有限元逼近。我们的构造依赖于Regge有限元，这是分段多项式对称（0，2）张量场具有单值切线-切线分量沿着元素界面。当用于离散的黎曼度量张量，这些分段多项式张量场不具有足够的规则性，以定义连接和曲率在经典意义上，但我们展示了如何使这些数量的意义上的分布意义。然后，我们表明，这些分布量收敛在一定的对偶Sobolev规范下细化的三角剖分，他们的顺利同行。我们还讨论了分布曲率和分布联络在分片多项式有限元空间上的投影。我们表明，相关的投影算子与某些线性化微分算子交换，得到一个交换图的微分复形。
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.