An analysis of monotonicity conditions in the mixed hybrid finite element method on unstructured triangulations

In this paper we consider the mixed hybrid finite element method on unstructured triangular grids and evaluate its, monotonicity properties by using a non standard set of basis functions for the C velocity approximation space. The mixed hybrid discretization of the steady-state diffusion equation produces a system matrix that depends only on the inner product of the outward normals to the edges of the triangulation and not oil the choice of the velocity space basis. This property is used to study the characteristics of the system matrix. It is well known that this matrix is of type M if the angles of the triangulation are not bigger than pi/2. An M-matrix has a nonnegative inverse. i.e. all the elements are nonnegative. This implies the existence of a discrete maximum principle and thus monotonicity of the discretization. We show that, when the triangulation is of Delaunay type and satisfies the property that no circumcenters of boundary elements with Dirichlet conditions lie outside the domain, the inverse of the final matrix is always positive, even in the presence of obtuse angles.