Advanced Mathematical Methods
Functional Analysis and Numerics for PDEs
A.A. 2020/2021
Lecturer: M. Fedele
Laboratory 4
Stokes and Navier-Stokes Problems
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Exercise 1
Let us consider the following Stokes problem (Couette ow ), de ned on the rectangular domain
Ω = [0, 3] × [0, 1]:
−∆u + ∇p = 0, in Ω,
div u = 0, in Ω,
u = 0, for y = 0, 0 ≤ x ≤ 3,
(1)
u = 1 i + 0 j, for y = 1, 0 ≤ x ≤ 3,
∂ u − pn = 0,
for x = 0 and x = 3, 0 ≤ y ≤ 1.
∂n
1. Verify that u = [y, 0]> , p = 0 is the solution of the problem.
2. Write the weak formulation of problem (1).
3. Starting from lab4ex1_template.edp, implement in FreeFem++ the solution to problem
(1), on a mesh with elements size h = 0.1. Set the GMRES solver for the solution of the
corresponding algebraic system. Consider a stopping criterion with tolerance eps=10−8
and maximum number of iteration nbiter=10000.
4. Compute the solution of problem (1) with nite elements P1 /P0 , P1 /P1 , P1bubble /P1 , P2 /P1 .
Show that some of these couples produce spurious pressure modes.
5. Fill the table below with the results obtained at the previous point:
ku − uh kH 1 (Ω) kp − ph kL2 (Ω)
P1 /P0
P1 /P1
P1bubble /P1
P2 /P1
1
Exercise 2
Consider an incompressible Newtonian uid with kinematic viscosity ν = 0.02 m2 /s, in the duct
Ω = [0, 3] × [0, 1]. We denote with Γin the inlet boundary on the left, Γout the outlet boundary
on the right, and Γw the bottom and the top edges representing the wall of the duct. The ow
is governed by the following Navier-Stokes problem:
∂u
∂t + (u · ∇) u − ν∆u + ∇p = 0, in Ω, t ∈ (0, 3),
div u = 0, in Ω, t ∈ (0, 3),
u = g(x, y, t), on Γin , t ∈ (0, 3),
∂u (2)
ν − p n = 0, on Γout , t ∈ (0, 3),
∂n
u = 0,
on Γw , t ∈ (0, 3),
u = 0, for (x, y) ∈ Ω, t = 0,
where the inlet non-homogeneous Dirichlet condition is de ned as the following pulsatile parabolic
velocity pro le:
g(x, y, t) = 0.5 (cos(πt − π) + 1) [4y(1 − y) i + 0 j].
We notice that no-slip conditions (homogeneous Dirichlet) are imposed at the walls and zero-
traction conditions (homogeneous Neumann) are set at the out ow.
1. Write the weak formulation of problem (2); then discretize it with the implicit Euler method
and a semi-implicit treatment of the nonlinear term.
2. Starting from lab4ex2_template.edp, implement in FreeFem++ the solution of problem
(2) h = 0.1m and P1bubble /P1 nite elements.
Exercise 3
Let us consider an unsteady Stokes problem with viscosity ν = 0.05 in a rectangular cavity
Ω = [0, 3] × [0, 1], with free-slip conditions on the cavity walls:
∂u
∂t − ν∆u + ∇p = 0, for (x, y) ∈ Ω, t ∈ (0, 5),
div u = 0, for (x, y) ∈ Ω, t ∈ (0, 5),
−ν 1 = 0, u2 = 0, for y = 0, 0 ≤ x ≤ 3, t ∈ (0, 5),
∂u
∂y
u = 1 i + 0 j, for y = 1, 0 ≤ x ≤ 3, t ∈ (0, 5), (3)
for x = 3, 0 ≤ y ≤ 1, t ∈ (0, 5),
u1 = 0, ν ∂u ∂x = 0,
2
for x = 0, 0 ≤ y ≤ 1, t ∈ (0, 5),
∂u2
u1 = 0, −ν ∂x = 0,
u = 0, for (x, y) ∈ Ω, t = 0.
1. Write the weak formulation of problem (3) and then discretize it with the implicit Euler
method.
2. Using FreeFem++, solve problem (1) with P1bubble /P1 nite elements on a mesh of size h = 0.1
and a time step ∆t = 0.1. Set the GMRES solver for the solution of a single timestep.
Consider a stopping criterion with tolerance eps=10−8 and maximum number of iteration
nbiter=10000.
3. Comment on the results.
