This paper presents a novel approach to the construction of the lowest order $H(\mathrm{curl})$ and $H(\mathrm{div})$ exponentially-fitted finite element spaces ${\mathcal{S}_{1^-}^{k}}~(k=1,2)$ on 3D simplicial mesh for corresponding convection-diffusion problems. It is noteworthy that this method not only facilitates the construction of the functions themselves but also provides corresponding discrete fluxes simultaneously. Utilizing this approach, we successfully establish a discrete convection-diffusion complex and employ a specialized weighted interpolation to establish a bridge between the continuous complex and the discrete complex, resulting in a coherent framework. Furthermore, we demonstrate the commutativity of the framework when the convection field is locally constant, along with the exactness of the discrete convection-diffusion complex. Consequently, these types of spaces can be directly employed to devise the corresponding discrete scheme through a Petrov-Galerkin method.

翻译：本文提出了一种新颖方法，用于在三维修正单纯形网格上构建针对相应对流扩散问题的最低阶$H(\mathrm{curl})$和$H(\mathrm{div})$指数拟合有限元空间${\mathcal{S}_{1^-}^{k}}~(k=1,2)$。值得注意的是，该方法不仅便于函数本身的构造，还能同时提供相应的离散通量。利用这一方法，我们成功建立了一个离散对流扩散复形，并采用特殊的加权插值在连续复形与离散复形之间架起桥梁，从而形成连贯的框架。此外，我们证明了当对流场局部恒定时该框架的交换性，以及离散对流扩散复形的正合性。因此，此类空间可直接用于通过Petrov-Galerkin方法设计相应的离散格式。