Studying a graduate-level Math paper (referenced by an engineering one which applies finite elements), and I came across this:
Consider the usual Sobolev spaces $H^m(\omega)$ ($\omega>0$), with norm $\|\cdot\|_{m,\omega}$, and the closed subspace $H^1_0(\omega)$ consisting of functions in $H^1(\omega)$ with zero trace on $\partial\omega$, and $L^2_0(\omega)$ consisting of functions in $L^2(\omega)$ with zero mean in $\omega$. The scalar product in $L^2(\omega)$ is denoted by $(\cdot,\cdot)_\omega$ and its norm by $\|\cdot\|_{0,\omega}$.
For a $d$-dimensional problem on domain $\Sigma$, assume that $a^e$ is an inner-product into $W \subset [H^1_0(\Sigma)]^d$ and that, endowed with this inner product, $W$ is a Hilbert space, and set $\|w\|_e = (a^e(w,w))^\frac12$.
We assume that we have the following continuity estimate for all $w \in W$: $$\|w_h\|^2_e \le \beta\|w_h\|^2_{1,\Sigma} \tag{1}$$ Then by a standard inverse estimate $$ \|w_h\|^2_e \le \frac{\beta C^2}{h^2}\|w_h\|^2_{0,\Sigma} \tag{2}$$
(Comment: $h$ is the spatial step-size for the finite-element grid)
Questions: I am not sure why the continuity estimate in (1) is applicable, or how it transposes into (2) using a standard inverse estimate. Also not sure what a standard inverse estimate is - and searches on the topic have not been successful.
More importantly, what I am trying to do here is to obtain the values of $\beta$ and $C$ and ensure that given my step-size $h$, my finite element analysis makes sense. How can I possibly calculate these values? Not sure how to proceed, but I think understanding (1) and (2) better may help. Thank you!