Homework 6. (Due March 17)
Math. 639
Problem 1. Show that the general additive preconditioner results in a (·, ·)
symmetric and positive definite preconditioner provided that Ri , for i =
1, . . . , l is symmetric with respect to (·, ·) and are spectrally equivalent
to A−1
i .
Problem 2. Show that the general multiplicative preconditioner results in a
(·, ·) symmetric and positive definite preconditioner provided that Ri , for i =
1, . . . , l is symmetric with respect to (·, ·) and are spectrally equivalent
to A−1
i .
Problem 3. This problem uses the notation of class, i.e., A : Vh → Vh , the
elliptic projector Pi : Vh → Si , etc. Consider Si = span{φi }, for i = 1, . . . , m
where φi is a finite element basis function. For f in Vh , let Fei = (f, φi ).
(a) Compute Pi A−1 f = αi φi by deriving an expression for αi in terms of
e and Fe (here Fej := (f, φj )).
the entries of the stiffness matrix A
(b) Set
m
X
B=
Pi A−1
i=1
and consider the iteration
Pm
xj+1 = xj + B(f − Axj ).
Let xj = i=1 cji φi . Using part (a) above, find an expression for the
e=L
e+D
e +U
e . (Hint:
vector cj+1 in terms of Fe, cj and the splitting A
A preconditioner is represented in the implementation as a
map from inner product data to coefficient data. Use part (a)
to find the matrix representation of this map.)
Problem 4. This problem is a continuation of the previous (same notation),
however, we now restrict to the case of a the finite element basis functions
coming from the one dimensional example.
(a) What is the constant n0 appearing in Assumption O1 in this case.
(b) It was proved in class that there are constants c0 and c1 not depending
on h so that
e · x ≤ c1 h−1 |x|2 .
c0 h|x|2 ≤ (Ax)
Use this to derive a bound for Cs (the stability constant associated with
the above subspaces).
1
2
(c) What is the bound on the condition number of BA implied by the theorem
for the additive preconditioner given in class?