a posteriori error estimation for skewed elements

Question

I'm working with error estimates for Poisson's equation of the form

$$\mathcal{E}^2_T = h_T^2\|-\Delta u - f\|_{L^2(T)} + \sum_{e\in \partial T} h_e\|n\cdot \nabla u\|_{L^2(e)}$$

where $T$ is an element and $h_T, h_e$ are representative sizes of the element volume and face, respectively.

For skewed elements, I'm unclear on what is used for $h_T$ in practice. I can imagine several options (radius of the largest circumscribable sphere/circle, $d$th root of the volume $|T|^{1/d}$, max edge length) for $h_T$ (though I believe $h_e$ is always just given to be the size of the face?).

Is there a preferred choice of for (slightly) skewed elements?

Answer 1

You need to trace where this term comes from in the derivation of these error estimates. In your case, you get this from an interpolation estimate of the form $\|z-I_h z\|_{L^2(e)} \le h^{?} \|\nabla^2 z\|_{L^2(T)}$, if I recall correctly. You'll need to look up what exactly the estimate is, what it involves, and where the transformation from reference to real cell/edge enters here. You will then find that $h$ is related to the norm of the mapping from reference to real cell. What it is here, I don't know, but it can be found out.