5
$\begingroup$

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.

  1. Is $L$ a self-adjoint operator?
  2. Show that $(u, u_x) = \frac{1}{2}(u(1)^2 - u(0)^2)$
  3. 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.
  4. Show that the bilinear form is coercive.
  5. 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.
  • 2
    Your statement of question 2 is incorrect. It should most likely be $(u, u_x)$. And you should specify that the Hilbert space inner product is the $L^2$ one.2011-09-18
  • 5
    Questions 1, 2, and 4 are completely "definition" questions. Please at the very least try to do those yourself, or if you really cannot figure it out, edit the question to include the appropriate definitions and which parts of the definitions you do not know how to check.2011-09-18

1 Answers 1