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arXiv · 2108.07308

Determinant of the finite volume Laplacian

Thomas Doehrman·David Glickenstein
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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.

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Thomas Doehrman, David Glickenstein. 2021-08-16. Determinant of the finite volume Laplacian. https://arxiv.org/abs/2108.07308

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