Determinant of the finite volume Laplacian

TL;DR

This paper explores the geometric properties of the finite volume Laplacian across all dimensions, relating its determinant to volume measures derived from simplex geometry, and discusses conditions for its negative semidefiniteness.

Contribution

It generalizes previous two-dimensional results to higher dimensions, providing a geometric description of the Laplacian determinant in relation to simplex volumes.

Findings

Laplacian determinant relates to simplex volume quantities

In many cases, the Laplacian is negative semidefinite with constant kernel

Generalizes 2D geometric Laplacian results to higher dimensions

Abstract

The finite volume Laplacian can be defined in all dimensions and is a natural way to approximate the operator on a simplicial mesh. In the most general setting, its definition with orthogonal duals may require that not all volumes are positive; an example is the case corresponding to two-dimensional finite elements on a non-Delaunay triangulation. Nonetheless, in many cases two- and three-dimensional Laplacians can be shown to be negative semidefinite with a kernel consisting of constants. This work generalizes work in two dimensions that gives a geometric description of the Laplacian determinant; in particular, it relates the Laplacian determinant on a simplex in any dimension to certain volume quantities derived from the simplex geometry.