I am reading a paper that deals with a certain finite element method.
We have the weighted Sobolev space $\mathcal{W}_{\mu}$ that consists of functions for which the norm
$\Vert u \Vert_{\mu}^{2} = \int_{\mathbb{R}^{+}} \omega_{\mu}^{2}(x) u^{2}(x)dx + \int_{\mathbb{R}^{+}} \omega_{\mu}^{2}(x) x^{2} (u'(x))^{2}dx $
is finite. We then introduce the inner product and bilinear form and semi norm
$(u,v)_{\mu} = \int_{\mathbb{R}^{+}} \omega_{\mu}^{2}(x) u(x)v(x)dx$
$A_{\mu}(u,v) = C \int_{\mathbb{R}^{+}} \omega_{\mu}^{2}(x) x^{2} u'(x)v'(x)dx + \int_{\mathbb{R}^{+}} C(x) \omega_{\mu}^{2}(x) x u'(x)v(x)dx + D \int_{\mathbb{R}^{+}} \omega_{\mu}^{2}(x) u(x)v(x)dx$, where $C$ and $D$ are constants (positive) and $C(X)$ is some bounded function of $x$.
The bilinear form $A_{\mu}$ satisfies the well known continuity and coercivity conditions, with constants $\gamma$ and $\alpha$ respectively.
$\vert u \vert_{\mu}^{2} = \int_{\mathbb{R}^{+}} \omega_{\mu}^{2}(x) x^{2} (u'(x))^{2}dx$
We then have the weak formulation to find $V \in \mathcal{W}_{\mu}$ such that
$(V_{t},\phi)_{\mu} + A_{\mu}(V,\phi) = 0 \ \ \ \ \forall \phi \in \mathcal{W}_{\mu}$
and, for a finite dimensional subspace $\mathcal{W}_{h}$ of $\mathcal{W}_{\mu}$, the associated semi-discrete problem: Find $V_{h} \in \mathcal{W}_{h}$ such that
$((V_{h})_{t},\phi_{h})_{\mu} + A_{\mu}(V_{h},\phi_{h}) = 0 \ \ \ \ \forall \phi_{h} \in \mathcal{W}_{h}$ .
The author then introduces an "auxiliary backward problem": Find $v \in \mathcal{W}_{\mu}$ such that for $t>0$
$(\phi,v_{t})_{\mu} - A_{\mu}(\phi,v) = -(\phi,e_{h}) \ \ \ \ \forall \phi \in \mathcal{W}_{\mu}$, where $e_{h} = V -V_{h}$ and $\tau < t $
and its semi-discrete version:
$(\phi_{h},(v_{h})_{t})_{\mu} - A_{\mu}(\phi_{h},v_{h}) = -(\phi_{h},e_{h}) \ \ \ \ \forall \phi \in \mathcal{W}_{h} \ \ \tau< t $ .
If $\delta_{h} = v - v_{h}$, the author claims that standard energy arguments we have:
$\int_{0}^{t} \vert \delta_{h}'(\tau) \vert_{\mu}^{2} + \Vert \delta_{h}(\tau) \Vert_{\mu}^{2} d \tau \leq C \int_{0}^{t} \vert e_{h} \vert_{\mu}^{2} + \inf_{w_{h} \in \mathcal{W}_{h}}\int_{0}^{t} \Vert v - w_{h} \Vert_{\mu}^{2} d \tau $
I cannot for the life of me see how to derive this inequality and the author makes no references in this regard. Any advice on how to proceed, or references to similar work would be appreciated.