Are there $P0$ (zero degree/constant element) nonconforming methods for approximating solutions in $H1$? More specifically, I have the equation:
$$u-f - T\Delta u = 0$$
Which can be interpreted as heat diffusion for time $T$ with an implicit step in time. I know that this can easily be handled with CG (continuous Galerkin) or DG (discontinuous Galerkin) as long as I use elements with degree $\geq 1$. I would like to use $P0$ elements however. The main issue is that the gradient in the variational formulation vanishes for constant elements.
Finite differences in some sense achieve this on a regular grid but I have an arbitrary mesh. So references on discretizations of the Laplace operator on discontinuous piecewise constant meshes are also welcome (I mainly deal with the 2D setting).
I am not that familiar with those but maybe finite volume (cell-centered) methods can handle this. I've seen some schemes but those considered equations without a reaction term.