We present a new mimetic finite difference method for diffusion problems that converges on grids with \textit{curved} (i.e., non-planar) faces. Crucially, it gives a symmetric discrete problem that uses only one discrete unknown per curved face. The principle at the core of our construction is to abandon the standard definition of local consistency of mimetic finite difference methods. Instead, we exploit the novel and global concept of $P_{0}$-consistency. Numerical examples confirm the consistency and the optimal convergence rate of the proposed mimetic method for cubic grids with randomly perturbed nodes as well as grids with curved boundaries.

翻译：我们提出了一种新的适用于扩散问题的仿射有限差分法，该方法在具有弯曲（即非平面）面的网格上收敛。关键在于，它生成一个对称的离散问题，每个弯曲面仅使用一个离散未知量。我们构建的核心原则是放弃仿射有限差分法局部一致性的标准定义，转而采用新颖的全局概念——$P_{0}$一致性。数值算例验证了所提仿射方法在随机扰动节点的立方网格以及具有弯曲边界的网格上的一致性和最优收敛速率。