18.303 Midterm, Fall 2010
Each problem has equal weight. You have 55 minutes.
Problem 1: Finite differences (20 points)
From class and homework, the d2 /dx2 operator on [0, L] can be discretized into values um ≈ u(m∆x) at points
x = m∆x [for ∆x = L/(M + 1)] as A = −DT D/∆x2 , where




1
−1 1
 −1 1





−1 1




..
..
DNeumann = 
DDirichlet = 
,

.
.
.
.
..
..






−1 1
−1 1
−1
when the boundary conditions are Dirichlet u(0) = u(L) = 0 [D an (M +1)×M matrix] and Neumann u0 (0) = u0 (L) = 0
[D an (M − 1) × M matrix], respectively.
(a) Write down a new D matrix that implements the boundary conditions: u(0)+u0 (0) = 0, u(L) = 0. For simplicity,
apply the left boundary condition at ∆x/2 rather than at 0, using u(∆x/2) ≈ (u0 + u1 )/2. Be sure to indicate
how many rows and columns your D matrix has, and for what m values you have degrees of freedom um .
(b) If you use your D matrix from the previous part to solve u00 (x) = f (x) approximately via A = −DT D/∆x2 , how
fast would you expect the errors to vanish as ∆x → 0? [i.e. errors proportional to ∆xn for what power n?]
Problem 2: Adjoints and stuff (20 points)
Let Ω ⊆ R2 be the rectangular 2d region x ∈ [0, Lx ], y ∈ [0, Ly ], with Dirichlet boundaries u|dΩ = 0. Consider the
operator
∂
∂u
∂
∂u
2
Âu = ∇ u +
c(x, y)
+
c(x, y)
∂x
∂y
∂y
∂x
h
i
∂
∂
∂
for some real-valued function c(x, y). Let B̂ = ∂x
c(x, y) ∂u
c ∂y
, i.e. B̂u = ∂x
∂y , the second term in Â.
R
RL RL
(a) Find B̂ ∗ =_____________ under the inner product hu, vi = Ω uv = 0 x 0 y dx dy u(x, y) v(x, y) [use
real-valued functions for simplicity]. Hence conclude that Â∗ =________.
∂u
∂t
have solutions u(x, y, t) that → 0 as t → ∞ for any initial
condition
1 c
u(x, y, 0)? Hint: consider hu, Âui for u 6= 0, and note that the eigenvalues of the 2 × 2 matrix
are
c 1
R
2
1 ± c. (From class: hu, ∇ ui = − Ω ∇u · ∇u.)
(b) Under what conditions on c(x, y) will Âu =
Problem 3: Thinking Green (20 points)
Consider the operator  = −c(x)∇2 in some 2d region Ω ⊆ R2 with Dirichlet boundaries (u|dΩ = 0), where c(x) > 0.
Suppose the eigenfunctions of  are un (x) with eigenvalues λn [that is, Âun = λn un ] for n = 1, 2, . . ., numbered in
order λ1 < λ2 < λ3 < · · · . Let G(x, x0 ) be the Green’s function of Â.
P
(a) If f (x) = R n αn un (x) for some coefficients αn =_________________ (expression in terms of f and
un ), then Ω G(x, x0 )f (x0 )d2 x0 =__________________ (in terms of αn and un ).
(b) The maximum possible value of
1
u(x)G(x, x0 )u(x0 ) d2 x d2 x0
Ω Ω c(x)
,
R |u(x00 )|2
d2 x00
Ω c(x00 )
R R
for any possible u(x), is _____________________ (in terms of quantities mentioned above). [Hint:
min–max.]
1