### Lectures:

Lectures Monday 10-12 and Thusday 8-10 in Kollektorn as usual.

Content: Variational formulation of Poisson's equation. Finite element methods. Read Chapter 8 and 15 of CDE or Fem in 1d, Fem in 23d

### Exercises:

Recommended, where Lx:y = exercise y under lecture x, E = exercise (pdef2-blandade övn), P = problem (Övningsexempel i PDE):

Demo:

For the solution u of the heat equation with u_0=0 and f a unit mass point load at x=0, compute lim_{t->oo}u(x,t). Compare to the fundamental solution of Poissons equation.
Show that for a solution to Poissons eq with homogeneous Dirichlet boundary conditions one has F(u)=-1/2\int_\Omega fu.
L5:1,2.

Individual work:

L4:1 compute c in fundamental solution.
L4:2 sol of non-homogeneous heat eq with unit mass load at x=0 and t=0.
L4:4 sol of non-homogeneous wave eq
L4:5 sol of wave eq with c^2.
L4:7 wave eq with boundary condition u(0,t)=0.
L4:8,9 the exploding ballon problem
P: 3, 6, 23
L5:3.
u=x^2+y^2-1 is the solution of a Poissons eq with homogeneous Dirichlet boundary conditions on the unit disc in 2D. Compute the total energy F(u) of the system. Consider some other function v satisfying the same boundary conditions, and verify that indeed F(v)>F(u).

/Claes