Mathematics homework help. MATH572 Spring 2020
Assignment 3
Due: Wednesday March 11
Solve any set of problems for 100 points.
Problem 1: (30 points) Let Ω ⊂ R
2
and u ∈ H1
(Ω). Prove the following inequality
Z
∂Ω
u
2
dx ≤ Ckuk
2
1
(1)
Z

u
2
dx ≤ C
Z

|∇u|
2
dx +
Z
∂Ω
u
2
dx
(2)
Problem 2: (30 points) Consider the following B.V.P for elliptic equation in R
2
.
−∆u + q(x)u = f(x, y), (x, y) ∈ Ω = (0, 2) × (0, 1)
and
∂u
∂ν = g, (x, y) ∈ Γ,
where Γ is its boundary, and ν is the outward unit normal vector to Γ. Here q = 1 in (0, 1) × (0, 1)
and q = 0 in the remaining part of the domain.
Derive the weak formulation for this problem and show the coercivety of the corresponding bilinear form in H(Ω)− norm.
Problem 3: (50 points) Consider the τ to be a tetrahedron in the (x, y, x)− space determined
by its vertexes P1, P2, P3, P4. In order these four points to form a tetrahedron we assume that they
are not in a plane . Let Σ = {v(P1), v(P2), v(P3), v(P4)} be the set of values of a function v at
the vertices. Find a nodal basis for the space of linear functions over τ by using homogeneous
(baracentric) coordinates (λ1, λ2, λ3, λ4). Compute the element mass matrix.
Problem 4: (20 points) Let Ω be the square (0, 1) × (0, 1). Prove the Poincare inequality
kuk
2
L2(Ω) ≤ C
k∇uk
2
L2(Ω) +
Z

u dx2
!
.
Problem 5: (20 points) Let τ be a shape regular square in 2 − D with a side hτ . If ∂τ is the
1
boundary of τ show that there is a constant C independent of hτ such that
kvk
2
L2(∂τ) ≤ C

h
−1
τ kvk
2
L2(τ) + hτk∇vk
2
L2(τ)

.
Problem 6: (20 points) Consider (τ,P, Σ), where
τ = {rectangle (xi−1, xi) × (yj−1, yj ) with vertices P1, P2, P3, P4};
P = {v : v(x, y) = a00 + a10x + a01y + a11xy + a20x
2 + a21x
2
y + a12xy2 + a02y
2
};
Σ = {v(P1), v(P2), v(P3), v(P4), v(P12), v(P23), v(P34), v(P41)}
where Pij is the mid point of the edge joining Pi and Pj
. Show that the set Σ is P− unisolvent.
Note that the term x
2
y
2
is missing in the polynomial set and the center of the rectangle is allso
missing from the set of points values so that dimP = 8 and the number of degrees of freedom is 8

Mathematics homework help