Consider the elliptic problem $Lu = \exp(x)$ on $[0,1]$ with $Lu = -\frac{d^2u}{dx^2} + \frac{du}{dx}$ and boundary conditions $u(0) = 5$, $\frac{du}{dx}(1) + u(1) = 2$. Answer the following questions.
- Is $L$ a self-adjoint operator?
- Show that $(u, u_x) = \frac{1}{2}(u(1)^2 - u(0)^2)$
- Put the system in the form $a(w,u)=F(w)$ and give both $a(w, u)$ and $F(w)$ such that we successfully can show coercivity of the bilinear form.
- Show that the bilinear form is coercive.
- Let a linear function space in $C^2[0, 1]$ be spanned by $\{Q_1(x), Q_2(x), ..., Q_N(x)\}$ with $Q_i(0) = 0; i = 1, ...,N$. Give the linear system that arises from the Galerkin projection of the above problem on this space.