Finite element approximation of the diffusion operator on tetrahedra

Linear Galerkin finite element discretizations of the Laplace operator produce nonpositive stiffness coefficients for internal element edges of two-dimensional Delaunay triangulations. This property, also called the positive transmissibility (PT) condition, is a prerequisite for the existence of an M-matrix and ensures that nonphysical local extrema are not present in the solution. For tetrahedral elements, it has already been shown that the linear Galerkin approach does not in general satisfy the PT condition. We propose a modification of the three-dimensional Galerkin scheme that, if a Delaunay triangulation is used, satisfies the PT condition for internal edges and, if further conditions on the boundary are specified, yields an M-matrix. The proposed approach can also be extended to the general diffusion operator with nonconstant or anisotropic coefficients.