18.086 Pset #4
18.086 - Spring 2016, Problem Set 3
Due before class starts on April 29. Late homework will not be given any credit.
Indicate on your report whether you have collaborated with others and whom
you have collaborated with.
1. (20 points)
a Problem 3 in Problem Set 7.2 of the textbook (eigenvalues of Gauss-Seidel).
April 9, 2016
b Problem 10 in Problem Set 7.2 of the textbook (spectral radius of red-black GaussSeidel).
2. (509points) Consider the domain ⌦, as given in the figure. The boundary
Out: Apr.
Dirichlet boundaries D (thin lines) and Neumann boundaries N (fat lines).
Due: May
2 we would like to approximate the Poisson problem
di↵erences,
8
<
4 u = 1 in
consists of
Using finite
⌦
Problem 1 (50 points) As in theu =last
f problem
on D , set, we consider the domain Ω
: @u
@f
= @n
on conditions
N
(see figure below), with Dirichlet @nboundary
at boundaries ΓD (thin
1
3
3
2
lines) and
conditions
parts
lines). yields
N (bold
whereNeumann
f (x, y) = x y boundary
xy 2 x . Write
a code that, at
for any
givenΓresolution
parameter,
Figure
In contrast to the last problem
set,0.1:
weDomain.
now solve for the steady state solution
of the diffusion
equation,
i.e. the
theabove
Poisson
a linear system
that discretizes
Poisson problem
problem. Then choose a small resolution,
a medium resolution and a high resolution, and (try to) solve the linear systems by
1. Matlab backslash
2.
3.
−∆u = 1 in Ω
u = f on ΓD
Jacobi iteration
∂u
∂f
a simple multigrid method (V-cycle)
=
on ΓN
∂n
∂n
(1)
(2)
(3)
1
where f (x, y) = x3 y − xy 3 − 21 x2 as before. Using finite differences, adapt your
code from the last pset in such a way that you can choose a resolution parameter
for the discretization (for instance the number of grid points per unit length).
Then choose a small resolution, a medium resolution and a high resolution, and
(try to) solve the linear systems by
• Matlab backslash (or any other built-in method in your favorite numerical
software)
• Jacobi iterations
1
• a simple multigrid v-cycle (see mit18086 multigrid.m for a 1D implementation)
• Conjugate gradient method (see mit18086 cg.m for a 2D Poisson CG
solver)
Compare the approaches with respect to accuracy and run times. Up to
which resolution can you go with the best method? Remarks/Hints:
• Use sparse matrices and reuse your codes from the last problem set.
• Codes from the CSE web page can be used.
• Be aware of the difference between discretization error and errors in solving
the linear system approximately.
• For a small test problem Au = b, the accuracy of a linear solver is best
measured by the norm of the error |utrue − uapprox |, where utrue can be
obtained by Matlab’s backslash operator or similar approaches in Mathematica etc. However, for high resolutions, this error may not be accessible.
In those cases, the residual |uapprox − b| can be used.
• The high resolution may cause some solution methods to fail. The medium
resolution shall be chosen in such a way that all considered
solution
methAbdulaziz
Albaiz
ods succeed to allow for comparison.
ional Science and Engineering II
Page 4 of 4
• The steady-state solution u should look like the figure below:
m is using several methods, and in each method, different grid sizes are used. The following
mmarizes all the combinations with their error (defined as ‖𝐴𝐴𝑥𝑥 − 𝑏𝑏‖∞) and run times (tolerance
7
Matlab’s
Backslash
Time: 0
nts)
5
Time: 0
nts)
1
Jacobi method
Time: 0
Gauss-Seidel
method
2
Conjugate
Multigrid
Gradient
(V-Cycle)*
Time: 0
Time: 0
Time: 0
Time: 0
Iterations: 233
Iterations: 185
Iterations: 79
Iterations: 0
Time: 0.04 sec
Time: 0.03 sec
Time: 0.03 sec
Time: 0.03 sec
Iterations: 1434
Iterations: 737
Iterations: 161
Iterations: 112
Time: 2.12 sec
Time: 0.9 sec
Time: 0.14 sec
Time: 0.67 sec
(base case)