Determinant of the Finite Volume Laplacian
Determinant of the Finite Volume Laplacian
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DOI：
10.1007/s00454-022-00429-1
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发表时间：
2021-08
影响因子：
0.8
通讯作者：
Thomas Doehrman;David Glickenstein
中科院分区：
文献类型：
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作者：
Thomas Doehrman;David Glickenstein
有限体积拉普拉斯可以在所有维度上定义，是在单纯网格上逼近算子的一种自然方式。在最一般的情况下，用正交对偶定义可能要求不是所有体积都是正的；例如，对应于非Delaunay三角剖分上的二维有限元的情况。然而，在许多情况下，二维和三维拉普拉斯可以被证明是负半定的，其核由常量组成。这项工作推广了二维的工作，给出了拉普拉斯行列式的几何描述；具体地说，它将任意维单纯形上的拉普拉斯行列式与从单纯形几何导出的某些体积量联系起来。
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.