MATH 6590, Homework 1
1. (6 pts) Let = (0, 1) and v H 2 (). Dene a grid on by 0 = x0 < x1 < < xn = 1 and
denote h = maxn (xi xi1 ). Let vI be the nodal value interpolation of v using continuous
i=1
piecewise linear polynomials. Prove that there exists a p
MATH 6590, Homework 2
1. (5 points) Textbook, p.67, problem 2.x.15.
2. (15 points) Let (H, (, ) be a Hilbert space with norm v H = (v, v), and V be a closed,
convex subset of H. Here V is convex means that (1 t)v1 + tv2 V for all v1 , v2 in V and
0 t 1. C
basisfunctions := matrix(3,1,
[ (xh/2)*(xh) / (0h/2) / (0h),
(x0)*(xh) / (h/20) / (h/2h),
(x0)*(xh/2) / (h0) / (hh/2) ]);
A := matrix(3,3):
for i from 1 to 3 do
for j from 1 to 3 do
A [i,j] := simplify(int(
diff(basisfunctions[i,1],x)
* diff(basisfunction
MATH 6590, Homework 5
1. (5 points) Write the weak formulation for the following problem:
u(4) = f in = (0, 1)
u(0) = 0, u (0) = 2
u (1) = 0, u (1) = 1
2. (15 points) Textbook 10.x.25. (|v|H 1 () in part (c) should be |H 1 () . You can use all results
v
p
MATH 6590, Homework 2
1. (5 points) Let = (a, b) (c, d) be a rectangular domain.
(a) Prove that for all v C0 (), there exists a constant c > 0 such that
(v, v) c v 2 ,
0
x
(1)
y
v
v
by using v(x, y) = a x (, y) d or v(x, y) = c y (x, ) d, and the Schwarz
MATH 6590, Homework 4
1. (10 points) Let be a polygon and Th be a quasi-uniform triangular mesh in satisfying the
minimum angle condition. Denote xi , i = 1, . . . , k to be the vertices in Th . We assume that
each vertex is shared by at most N triangles.