We present a simple and concise discretization of the covariant derivative vector Dirichlet energy for triangle meshes in 3D using Crouzeix‐Raviart finite elements. The discretization is based on linear discontinuous Galerkin elements, and is simple to implement, without compromising on quality: there are two degrees of freedom for each mesh edge, and the sparse Dirichlet energy matrix can be constructed in a single pass over all triangles using a short formula that only depends on the edge lengths, reminiscent of the scalar cotangent Laplacian. Our vector Dirichlet energy discretization can be used in a variety of applications, such as the calculation of Killing fields, parallel transport of vectors, and smooth vector field design. Experiments suggest convergence and suitability for applications similar to other discretizations of the vector Dirichlet energy.

中文翻译：

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矢量狄利克雷能量的简单离散化

我们使用 Crouzeix-Raviart 有限元对 3D 三角形网格的协变导数向量狄利克雷能量进行了简单而简洁的离散化。离散化基于线性不连续 Galerkin 元素，并且易于实现，不会影响质量：每个网格边有两个自由度，并且可以在所有三角形上一次性构建稀疏 Dirichlet 能量矩阵，使用仅取决于边长的简短公式，让人想起标量余切拉普拉斯算子。我们的向量狄利克雷能量离散化可用于多种应用，例如杀死场的计算、向量的平行传输和平滑的向量场设计。