# positivity preservation for mixed finite element discretization

I'm interested in mixed discretizations of the diffusion (and related) equations:

\begin{align} \frac{\partial h}{\partial t} + \nabla\cdot \mathbf q & = f \\ k^{-1}\mathbf q + \nabla h & = 0 \end{align}

where $$h$$ is a scalar field representing thickness or hydraulic head and $$\mathbf q$$ is a vector field representing flux or velocity. The primal form of this problem is obtained by eliminating $$\mathbf q$$, giving a second-order PDE for $$h$$. When discretizing the primal form with degree-1 continuous Galerkin basis functions, one can prove that, when the triangulation is nice, the resulting matrices are M-matrices and thus the problem has a discrete maximum principle. See for example the Kuzmin book.

Is the maximum principle still true with some mixed discretizations? For example, the MINI element $$CG_1 \times (CG_1 + B_3)^2$$ is inf-sup stable for mixed Poisson; does it have a discrete maximum principle for the thickness? What about DG $$\times$$ Raviart-Thomas (or some other hipster element) for the flux? The Godunov theorem tells us that there can't be a discrete maximum principle for higher-order basis functions, but there are still low-order bases for which it isn't necessarily an obstruction. Does the analysis change appreciably if the constitutive relation is nonlinear?