Could I apply a Q0-discretization to, say, the Poisson equation $\Delta \phi = f$ (where by Q0 I mean piecewise constant, and thus non-continuous, elements)?
Solving this with FEM, at least as I know it, involves the gradients of the discretized function $\nabla\phi$, which in a piecewise constant discretization I would expect to be zero almost everywhere, and undefined/infinity on element edges/surfaces.
Now what I'm not sure about is if using Q0 elements together with gradients simply doesn't make sense. I could probably figure out some kind of "workaround" like calculating a "pseudo-gradient" from the value differences to neighboring cells. But I suspect this would be equivalent to using a different type of finite element altogether. Maybe anyone can give me some insight on this?