Explicit geometric construction of sparse inverse mass matrices for arbitrary tetrahedral grids

TL;DR

This paper introduces a geometric method to construct sparse inverse mass matrices for tetrahedral grids, enabling efficient and explicit inversion crucial for finite element discretizations, even with anisotropic materials.

Contribution

It presents a unified geometric framework for constructing sparse inverse mass matrices for arbitrary tetrahedral grids, challenging the belief that barycentric dual grids cannot produce sparse inverses.

Findings

Sparse inverse mass matrices are explicitly constructed without inverting local matrices.

The new method accelerates computations and simplifies implementation.

Application to 3D Poisson problem demonstrates effectiveness.

Abstract

The geometric reinterpretation of the Finite Element Method (FEM) shows that Raviart Thomas and Nedelec 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 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 term. A major difference with the approaches proposed so far is that the consistent term 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.