A Simple Discretization of the Vector Dirichlet Energy

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.