MANDATORY EXERCISE
Everyone is encouraged to define their own project associated with
their master- or PhD-projects. The projects must however be approved. The topic must involve testing of a scheme for appropriate
numerical behaviour. PhD students (taking Mek9250) will need
to present their work by the end of the semester.
Exercise 1
Consider the following problem on the domain Ω = (0, 1)2 :
−∆u = f in Ω,
u = 0 for x = 0 and x = 1,
∂u
(3)
= 0 for y = 0 and y = 1.
∂n
Assume u = sin(πkx) cos(πky) and compute f = −∆u.
a) Compute analytically the H p norm of u.
b) Compute the L2 and H 1 errors for 1/h = 8, 16, 32, 64 when using first
and second order Lagrangian elements when k = 1, 10.
c) Consider the following error estimates
(1)
(2)
ku − uh k1 ≤ Cα hα
and
ku − uh k0 ≤ Cβ hβ
Estimate Cα , Cβ , α and β and check whether the expected error estimate is
valid. (Advice: since Ck hk is a polynom of order k on should check k dependence first. It is a polynom and hence a log-log plot will confirm whether it
is a good estimate or not as polynomial behaviour should appear as straight
lines. If the log-log plot demonstrate a straight line then you may use the
the least square method in the estimation.)
Exercise 2
Consider the following equation on the domain Ω = (0, 1)2 :
−µ∆u + ux = 0 in Ω,
u = 0 for x = 0,
u = 1 for x = 1,
∂u
(7)
= 0 for y = 0 and y = 1
∂n
a) Derive an expression for the analytical solution.
b) Compute the numerical error for µ = 1, 0.1, 0.01 at h = 8, 16, 32, 64.
c) Compare against the expected error estimate, which is on the same form
as in Exercise 1 and discuss the result.
(4)
(5)
(6)
1
2
MANDATORY EXERCISE
d) Implement the Streamwise Upwinding Petrov-Galerkin (SUPG) method
and compare against the results in b) and c).
Deadline: March 6. Please include code. Typesetting in LATEX is
prefered.