Quiz V, Math 574, 2 May, 2013
Name:
Solutions
1. (20 points): Prove that there is some constant C > 0, independent of h, such that
u uh
H1
0
+ p ph
L2
C
u vh
H1
0
+ p qh
,
L2
for arbitrary vh Xh and qh Qh , where (u, p) X Q and (uh , ph ) Xh Qh
are the so
Quiz II, Math 574, 28 February, 2013
Name:
Solutions
1. (10 points): Let K be an arbitrary triangle in x space. Suppose K is the reference
triangle in x space. Let FK : K K be the orientation preserving ane mapping
1
given by FK ( ) = a + BK x. Show that
Homework 08, Math 574
7 March, 2013
Name:
1. Suppose = (0, 1) (0, 1). Show that there exists a constant C > 0 such that
u
H2
C u
L2
,
2
for all u H0 ().
2. Consider the problem
d4 u
= f in 0 < x < 1,
dx4
u(0) = u (0) = 0 = u (1) = u (1).
(a) Write this i
Quiz IV, Math 574, 18 April, 2013
Name:
Solutions
1. (10 points): Set s := T /K , where K is a positive integer. Consider the CrankNicolson-Galerkin method for the diusion problem: given uk1 Vh , with 1
h
k K , nd uk Vh such that
h
uk uk1
1
h
h
, +
s
2
u
Homework 13, Math 574
18 April, 2013
Name:
Suppose that Xh X := H1 () and Qh Q := L2 () form a stable pair, i.e., the
0
0
(BBh ) condition is satised: there exists a > 0, such that
sup
0=vh Xh
(qh , vh )
qh
vh H1
L2
,
0
for all qh Qh . Suppose that (u, p
Quiz I, Math 574, 7 February, 2013
Name:
Solutions
1
1. (10 points): Let = (0, 1) and x y . Dene y : H0 () R via
y (f ) = f (y ).
1
(a) Show that y (H0 () .
1
(b) By the Riesz Representation Theorem, there exists a unique u H0 (), such
that
du dv
1
y (v )
Quiz III, Math 574, 21 March, 2013
Name:
1. (10 points): Suppose = (0, 1) (0, 1). Show that there exists a constant C > 0
such that
u H 2 C u L2 ,
2
for all u H0 ().
Solution: First, observe that, by denition,
u
2
H2
=u
2
L2
2
L2
+ x u
+ y u
2
L2
+ xx u
2
Programming Exercise, Math 574
24 January, 2013
Name:
1. Use the nite element method to approximate the solution of the following Poisson problem:
u = f ,
in
:= (0, 1) (0, 1) ,
u=0,
on
,
where f L2 () is given. You should respect the following guideline
Homework 11, Math 574
4 April, 2013
Name:
1. Set s := T /K , where K is a positive integer. Consider the Crank-Nicolson-Galerkin
method for the diusion problem: given uk1 Vh = M0,r , with 1 k K , nd
h
uk Vh such that
h
uk uk1
1
h
h
, +
s
2
uk + uk1 , = f
Homework 10, Math 574
21 March, 2013
Name:
1. Let R2 be open and polygonal, assume T > 0, and dene T := (0, T ).
Consider the diusion problem
t u(x, t) u(x, t) = f (x, t)
u(x, t) = 0
u(x, 0) = v (x)
(x, t) T ,
(x, t) (0, T ),
x .
(a) Write this problem in
Homework 07, Math 574
28 February, 2013
Name:
1. Let R2 be polygonal. A family of triangulations of , cfw_Th h , is said to be
globally quasi-uniform if and only if there exists a constant > 0, independent of
h, such that
h = max hK
K Th
.
1
min hK
K Th
Homework 05, Math 574
7 February, 2013
Name:
1. Let K be a triangle with vertex nodes ai , i = 1, 2, 3, numbered in counter-clockwise
order. Let i , i = 1, 2, 3, be the barycentric coordinates.
(a) Consider the points (1 , 2 , 3 ), where the i cfw_0, 1/3,
Homework 04, Math 574
7 February, 2013
Name:
1. Let = (0, 1) and let Th = cfw_K1 , . . . , KM +1 be a partition of , where Ki :=
[xi1 , xi ], and hi := xi xi1 , i = 1, . . . , M + 1. Dene
M0,r := v C 0 () v |Ki Pr (Ki ), i = 1, . . . , M + 1, and v (0) =
Homework 02, Math 574
24 January, 2013
Name:
1. Let V be a Hilbert space, and Vh , a nite dimensional subspace of V . Suppose
a( , ) : V V R is a symmetric, coercive, and continuous bilinear form, with
coercivity constant > 0 and continuity constant > 0.
Homework 01, Math 574
17 January, 2013
Name:
1. Let (V, ( , ) be an inner product space. Show that, for any u, v V ,
|(u, v )|
(u, u)
(v, v ),
with equality if and only if u = v , for some R.
2. Let (V, ( , ) be an inner product space. Prove that v
norm