Numerics for PDEs
FB 3 — Mathematisches Institut
Prof. Dr. Thomas Götz
MSc. Karunia Putra Wijaya
Exercise Sheet 7
29. Mai 2017
Exercise 25: Consider the Helmholtz equation −∆u + ω 2 u = f in a bounded domain Ω ⊂ Rn
with a Neumann boundary condition ∂n u = g on ∂Ω. Derive the weak form of the equation.
Hint: use v ∈ H01 as the test function.
Exercise 26: For the Laplace equation −∆u = f in Ω with the homogeneous Dirichlet boundary condition u = 0 on ∂Ω the finite element approximation leads to a system Ay = b where
Aij = h∇vi , ∇vj i,
for basis functions vi ∈ V ⊂ H01
bi = hf, vi i.
Show that the system matrix A is
(a) symmetric and
(b) strictly positive definite.
What can you say about the solvability of Ay = b?
Exercise 27: Consider the 1D Poisson equation −u00 = f on the unit interval Ω = (0, 1) with
homogeneous theDirichlet boundary conditions. Show that for every source f , the finite element
method using a single Lagrange–element with the three nodes x0 = 0, x ∈ Ω arbitrary and
x2 = 1 computes the correct function value u(x).
Exercise 28:
Consider the 1D problem −u00 + u = 10x in Ω = (0, 1) where u(0) = u(1) = 0.
(a) Derive the weak form of the problem.
(b) Compute the system matrix A and load vector b for the finite element approximation using
the linear Lagrange basis function


4(x − xi−1 ), x ∈ [xi−1 , xi ]
vi (x) = 4(xi+1 − x), x ∈ [xi , xi+1 ]


0,
else
where xj = j/4 for j = 0, · · · , 4 (5 grid-points).
(c) Calculate the approximate solution y of the problem by solving Ay = b.
(d) Calculate the analytic solution u of the problem.
Hint: use the variation of constants known from some ODE lectures.
Discussion in week number 24
1/1