The geometric reinterpretation of the Finite Element Method (FEM) shows that Raviart–Thomas and Nédélec mass matrices map from degrees of freedoms (DoFs) attached to geometric elements of a tetrahedral grid to DoFs attached to the barycentric dual grid. The algebraic inverses of the mass matrices map DoFs attached to the barycentric dual grid back to DoFs attached to the corresponding primal tetrahedral grid, but they are of limited practical use since they are dense.

In this paper we present a new geometric construction of sparse inverse mass matrices for arbitrary tetrahedral grids and possibly inhomogeneous and anisotropic materials, debunking the conventional wisdom that the barycentric dual grid prohibits a sparse representation for inverse mass matrices. In particular, we provide a unified framework for the construction of both edge and face mass matrices and their sparse inverses. Such a unifying principle relies on novel geometric reconstruction formulas, from which, according to a well-established design strategy, local mass matrices are constructed as the sum of a consistent and a stabilization part. A major difference with the approaches proposed so far is that the consistent part is defined geometrically and explicitly, that is, without the necessity of computing the inverses of local matrices. This provides a sensible speedup and an easier implementation. We use these new sparse inverse mass matrices to discretize a three-dimensional Poisson problem, providing the comparison between the results obtained by various formulations on a benchmark problem with analytical solution.

有限元方法（FEM）的几何重新解释表明，Raviart-Thomas和Nédélec质量矩阵从附着于四面体网格几何元素的自由度（DoF）映射到附着于重心双网格的DoF。质量矩阵的代数逆将映射到重心双网格的DoF映射回绑定到相应的原始四面体网格的DoF，但是由于它们很密集，因此在实际应用中受到限制。

在本文中，我们提出了稀疏逆质量矩阵的一种新的几何构造对于任意的四面体网格以及可能的不均匀和各向异性的材料，这打破了传统的常识，即重心双网格禁止对逆质量矩阵进行稀疏表示。特别是，我们为边缘和面质量矩阵及其稀疏逆的构造提供了一个统一的框架。这样的统一原理依赖于新颖的几何重构公式，根据一种公认的设计策略，可以根据该公式重新构造局部质量矩阵，将其作为一致部分和稳定部分的总和。与迄今为止提出的方法的主要区别在于，一致的部分是在几何上和明确定义的，也就是说，无需计算局部矩阵的逆。这提供了合理的加速和更容易的实现。