# $P0$ elements for $H1$

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.

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.

• Is there a result on the convergence for P0 elements? What flavor of DG would this work for (e.g. IPG, LDG, etc.)? – lightxbulb Jul 1 at 18:51
• I have to admit that I don't know the literature well enough but would assume that for all of these, using $P_0$ elements converges. (Of course, convergence for these low-order elements might be unbearably and impractically slow, but that's a different issue.) – Wolfgang Bangerth Jul 2 at 1:36
• From what I have found in literature this seems to be dependent on the mesh, problem, and scheme. The main issue is that if we have $O(h^p)$ then this will not even converge theoretically for $p=0$. For instance NIPG has $O(h^p)$ convergence for even $p$ on arbitrary meshes for the Poisson equation, while SIPG has $O(h^{p+1})$. Some of the results I have found come from here: "Numerical Analysis of Higher Order Discontinuous Galerkin Finite Element Methods", R. Hartmann. – lightxbulb Jul 2 at 7:49
• In the case of ${\cal O}(h^p)$, when you have $p=0$, you generally do converge the error to zero, just not an any particular rate. In other words, you'll get $o(1)$ convergence, which means that the error goes to zero, but not at a rate that would follow a $h^\alpha$. – Wolfgang Bangerth Jul 3 at 16:51
• I had the impression that $O(1)$ will leave you with a constant error term that will remain regardless of the number of degrees of freedom. For example this behaviour can be seen in Table 4.1 of: researchgate.net/publication/… Am I misunderstanding something regarding DG's error bound?At least for FD schemes I know I can easily construct nonsensical schemes with $O(1)$ that do not converge. Is there a key difference that I am missing? – lightxbulb Jul 3 at 17:13

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 $$\sum\tau_f \int_f [u]v = \sum \tau_f |f| (u^+-u)$$, 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 $$f$$.

• What do you mean by "do reduce to TPFA"? Does TPFA match SIPG, NIPG, or IIPG for specific parameters? The mesh orthogonality is a requirement on TPFA. As far as I am aware there is no such requirement on interior penalty DG, but it can still converge in some cases. – lightxbulb Jul 18 at 1:00
• I think it matches all of them 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 $\sum \tau_f \int_{f} [u] v = \sum \tau_f |f| (u^+-u)$, which is similar to TPFA if $\tau_f$ is defined as the inverse distance from the two cell centers. – Jesse Chan Jul 18 at 14:28
• While this is interesting in its own right, it seems more like trying to fit one method to the other artificially, which is not really how things stand in practice (for me). For example one of the main reasons I will likely transition to SIPG is to allow for: 1) higher degrees in an uncomplicated manner, 2) not having the restrictive orthogonality requirement on the mesh. And even then SIPG would supposedly converge to the right solution, or at least that is what I am envisioning. – lightxbulb Jul 18 at 17:22
• Sure. I'm not trying to push TPFA (I actually work in high order DG myself), just trying to explain what you might expect for convergence if you use p=0 and SIPG. – Jesse Chan Jul 19 at 1:01