Numerical Methods for PDEs
MATH 610 - 600
Alan Demlow
Homework 3
Spring 2015
Homework 3
Exercise 1
50%
Let Ω ⊂ R2 be a bounded polygonal domain. Let f, g ∈ L2 (Ω) and consider the system for u and
φ
−∆u − φ = f
in
Ω,
u − ∆φ = g
in
Ω,
supplemented by the boundary condition u = φ = 0 on ∂Ω.
1. Consider the weak formulation of the above system : Seek u ∈ V, φ ∈ W such that
Z
Z
a((u, φ), (v, ψ)) =
f v+
g ψ,
∀(v, ψ) ∈ V × W.
Ω
Ω
Determine V, W and the bilinear form a(., .) such that there exists a unique solution (u, φ) ∈
V × W.
2. Prove using the Lax-Milgram theorem that the weak formulation indeed has a unique solution
and deduce a stability bound for the solution.
3. For a domain Ω = Ωd := (−d, d)2 , show that there exists an absolute constant c such that
kukL2 (Ω) 6 cdk∇ukL2 (Ω) ,
for any function u ∈ H01 (Ωd ).
4. Consider the modified problem
−∆u + φ = f
in
Ωd ,
u − ∆φ = g
in
Ωd .
Determine a new weak formulation and prove the existence and uniqueness of a solution
provided that d is small enough. Derive a stability bound.
Exercise 2
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Let K be a nondegenerate triangle in R2 . Let a1 , a2 , a3 be the three vertices of K. Let aij = aji
denote the midpoint of the segment (ai , aj ), i, j ∈ {1, 2, 3}. Let P2 be the set of polynomials over
K of total degree at most 2. Let Σ = {σ1 , σ2 , σ3 , σ12 , σ23 , σ31 } be the functionals (or degrees of
freedom) on P2 defined as
σi (p) = p(ai ), i ∈ {1, 2, 3}
σij (p) = p(ai ) + p(aj ) − 2p(aij ), i, j = 1, 2, 3, i 6= j.
(a) Show that Σ is a unisolvent set for P2 (this means that any p ∈ P2 is uniquely determined by
the values of the above degrees of freedom applied to p).
(b) Compute the nodal basis of P2 which corresponds to {σ1 , . . . , σ31 }.
(c) Evaluate the entry m11 of the element mass matrix.
Hint. If you have problem computing the functions from (b) for a general triangle, derive them for
a reference element.
Exercise 3
25%
Let K = [0, 1]2 be the unit square and denote by qi , i = 1, ..., 4, its vertices and by ai , i = 1, .., 4,
the midpoints of its sides. Let P = Q1 := {p(x, y) = (ax + b)(cy + d) : a, b, c, d ∈ R} be the space
of polynomials of degree at most 1 in each direction.
1. For Σ := {σ1 , σ2 , σ3 , σ4 }, where σi (p) = p(qi ), i = 1, ..., 4, show that the finite element triplet
(K, P, Σ) is unisolvent.
e := {e
2. For Σ
σ1 , σ
e2 , σ
e3 , σ
e4 }, where σ
ei (p) = p(ai ), i = 1, ..., 4, show that the finite element triplet
e is not unisolvent.
(K, P, Σ)