Let $V=H_0^1(0,1)$ and consider the bilinear form acting on $V$: $$a(u,v) = \int_0^1 \frac{1}{1+x} u'(x)v'(x) dx $$ and let $F:V \rightarrow \mathbb{R}$ the functional defined as $$F(v)=\int_0^1 \Bigl(-e^x - \frac{1}{(1+x)^2} \Bigr) v dx$$ Consider now the weak formulation $a(u,v)=F(v)$.
In Finite Element method context, if $V_h$ is a finite dimensional subspace of $V$ (usually $V_h$ is spanned by hat functions), I have the estimate, called Cèa Lemma, where $u \in V$, $u_h \in V_h$ (the approximate solution):
$$||u-u_h||_V \leq \frac M\alpha \inf_{w_h \in V_h} ||u-w_h||V$$ where $\alpha$ is the constant of coercivity, and $M$ the constant of the continuity.
I want to compute this special bound in this case, so I need to compute those constants.
- Coercivity:
$$a(u,u) \geq \frac12 \int_0^1 u'(x)^2 dx \geq \frac{1}{2(1+C_P)^2} ||u||_V^2$$ where $C_P$ is the Poincarè constant. Therefore I set $$\alpha=\frac{1}{2(1+C_P)^2}$$
- Constant of continuity:$|F(v)| \leq \int_0^1 |e^x + \frac{1}{(1+x)^2}| |v(x)|dx $
Now, for $x \in (0,1): e^x + \frac{1}{(1+x)^2} \leq e + 1$, therefore: $$|F(v)| \leq (e+1) \int_0^1 1\cdot |v(x)| dx \leq (e+1) ||v||_V$$ where I used the fact that $$||v||_V^2 = ||v||_{L^2}^2 + ||v'||_{L^2}^2$$
In conclusion $$||u-u_h||_V \leq \frac{2(e+1)}{1+C_P^2} \inf_{w_h \in V_h} ||u-w_h||_V$$