Non-negative mixed finite element formulations for a tensorial diffusion equation

We consider the tensorial diffusion equation, and address the discrete maximum–minimum principle of mixed finite element formulations. In particular, we address non-negative solutions (which is a special case of the maximum–minimum principle) of mixed finite element formulations. It is well-known that the classical finite element formulations (like the single-field Galerkin formulation, and Raviart–Thomas, variational multiscale, and Galerkin/least-squares mixed formulations) do not produce non-negative solutions (that is, they do not satisfy the discrete maximum–minimum principle) on arbitrary meshes and for strongly anisotropic diffusivity coefficients.