$P0$ elements for $H1$

Question

Are there $P0$ (zero degree/constant element) nonconforming methods for approximating solutions in $H1$? More specifically, I have the equation:
$$u-f - TDelta u = 0$$
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.

Jesse Chan · Accepted Answer

To expand on Wolfgang Bangerth's answer, I think P0 DG schemes reduce to two-point cell-centered finite volume schemes. I don't know if DG convergence analysis always includes $p = 0$, but the resulting finite volume schemes can be shown to converge under appropriate "mesh orthogonality" conditions.
https://math.unice.fr/~minjeaud/Donnees/JourneesNumeriques_14-1/TP/Nice2014.pdf
Edit: including comment in the main answer. I think TPFA is equivalent to all of the DG methods (SIPG, NIPG, IIPG) for an appropriately defined mesh-dependent penalty parameter. For example, assuming $u$ is constant and $v=1$, all terms in the for SIPG bilinear form drop out except for the penalty term $sumtau_f int_f [u]v = sum tau_f |f| (u^+-u)$, which is identical to TPFA if $tau_f$ is defined as the inverse distance from the cell centers of the elements connected to each face $f$.

Wolfgang Bangerth · Answer

You can use discontinuous Galerkin methods also for $P_0$ elements. It's true that the gradient in the cell interior is zero, so your formulation will exclusively consist of the jump terms at cell interfaces.

Answered by Wolfgang Bangerth on August 22, 2021

$P0$ elements for $H1$

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