Three-Dimensional Delaunay Triangulations for Finite Element Approximations to a Second-Order Diffusion Operator

Abstract

The second-order diffusion operator given by $\mathcal{L}u = \nabla \cdot ({\bf K}\nabla u)$ is studied in terms of finite element numerical solutions. When a standard Galerkin finite element approximation of the operator is arranged in a specific manner, a condition of positive connection values is imposed that is necessary to produce an M-matrix in a linear system when boundary nodes are properly handled. In two dimensions, the condition is achieved when a Delaunay triangulation is imposed. However, it is shown that a three-dimensional Delaunay triangulation does not generally produce a discretisation satisfying the condition. Further, it is generally not possible to produce a three-dimensional triangulation that satisfies the positive interior connection condition.