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?



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