Courses/FEM/modules/existence
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Contents
1 Existence and uniqueness of solutions
1.1 Precondition
1.2 Theory
1.2.1 Abstract framework for linear elliptic PDE
1.2.1.1 Linear form
1.2.1.2 Bilinear form
1.2.1.3 Existence and uniqueness
1.2.1.4 Function space
1.2.2 Lax­Milgram's theorem
1.2.2.1 Form requirements
1.2.3 Poincaré's inequality
1.2.4 Example equation
1.3 Software
1.4 Postcondition
1.5 Exercises
1.6 Examination
1.7 [TODO]
Existence and uniqueness of solutions
Precondition
Science
Function approximation
Theory
Abstract framework for linear elliptic PDE
Function space (Hilbert space) V with norm ∥v|V and inner product (v, w)V .
Bilinear form a
V × V → R
determined by underlying differential equation.
Linear form L
V → R
determined by data.
Search for solution in function space V , formulate equation using a and L.
Linear form
A linear form L(v) is a function taking values (functions) in V into R
L(v) ∈ R∀v ∈ V
. L(v) linear in the argument
L(α1 v1 + α2 v2 ) = α1 L(v1 ) + α2 L(v2 )
∀ αi ∈ R, vi , ∈ V
Bilinear form
A bilinear form a(v, w) is a function taking values (functions) in V
× V
into R
a(v, w) ∈ R∀v, w ∈ V
.
linear in each argument
a(v, w)
a(α1 v1 + α2 v2 , w 1 ) = α1 a(v1 , w 1 ) + α2 a(v2 , w 1 )
a(v1 , α1 w 1 + α2 w 2 ) = α1 a(v1 , w 1 ) + α2 a(v2 , w 1 )
∀ αi ∈ R, vi , w i ∈ V
Existence and uniqueness
Find u
∈ V
such that
a(u, v) = L(v)
Problem: do such solutions u
∈ V
exist? (Existence)
Problem: if so, is u unique? (Uniqueness)
∀v ∈ V
For certain requirements (a certain class of equations: linear and elliptic) on a(v, w), L(v) and V , we can
show existence and uniqueness of solutions u.
Function space
Natural space for second­order elliptic differential equations:
Bounded function and gradient
2
V = {v : ∫
(|∇v|
2
+ v )dx < C}
Ω
Inner product
(v, w )
V
= ∫
(∇v ⋅ ∇w + vw)dx
Ω
Generates the norm
−
−
−
−
−
−
−
−
−
−
−
−
−
−
∥v∥
V
= √∫
2
(|∇v|
2
+ v )dx
Ω
Usual notation
V = H
1
(Ω)
where index 1 denotes one derivative (H 2 would add a term with the second derivative). Named after
Sobolev.
Lax-Milgram's theorem
If a(v, w) is a continuous, coercive, bilinear form on V and L(v) is a continuous, linear form on V , then
there exists unique u ∈ V satisfying a(u, v) = L(v), ∀v ∈ V .
Form requirements
Coercivity (or V­ellipticity) of a(v, w):
There exists constant κ1
> 0
a(v, v) ≥ κ1 ∥v∥
Continuity of a(v, w):
2
V
There exists constant κ2
> 0
|a(v, w)| ≤ κ2 ∥v∥
V
∥w ∥
V
Continuity of L(v):
There exists constant κ3
> 0
|L(v)| ≤ κ3 ∥v∥
V
Discussion of continuity (by linearity)
|L(v)| ≤ κ3 ∥v∥
V
⇒ |L(v) − L(w)| ≤ κ3 ∥v − w ∥
V
Recall Lipschitz continuity for a function f
R → R
|f(x) − f(y)| ≤ L|x − y|
In this course we also require that a(v, w) be symmetric when discussing existence/uniqueness.
Poincaré's inequality
∥v∥
2
L2 (Ω)
for v =
≤ C(∥v∥
2
L2 (Γ)
+ ∥∇v∥
0, x ∈ Γ ⇒ ∥v∥
L2 (Γ)
2
L2 (Ω)
)
= 0
∥v∥
L2 (Ω)
≤ C∥∇v∥
L2 (Ω)
Consequence
∥v∥
L2 (Ω)
+ ∥∇v∥
L2 (Ω)
≤ C2 ∥∇v∥
Example equation
−Δu + u = f
∇u ⋅ n = 0,
x ∈ Γ
Linear form
L(v) = (f, v)
L2 (Ω)
Bilinear form
a(u, v) = (∇u, ∇v) + (u, v)
Verification of requirements for Lax­Milgram's theorem:
Coercivity
a(v, v) = (∇v, ∇v) + (v, v) = ∥∇v∥
2
+ ∥v∥
2
= ∥v∥
2
V
⇒
κ1 = 1
Continuity of a
|a(v, w)| = |(∇v, ∇w) + (v, w)| = |(v, w )
V
| ≤ ∥v∥
V
∥w ∥
V
⇒
κ2 = 1
Continuity of L
|L(v)| = |(f, v)| ≤ ∥f ∥
κ3 = ∥f ∥
L2
∥v∥
L2
≤ ∥f ∥
L2
∥v∥
V
⇒
L2
where we "for free" see the requirement that ∥f∥ L must be bounded to satisfy the requirements.
2
Software
Postcondition
You should now be familiar with:
Lax­Milgram's theorem
Poincaré's inequality
Standard function space V and norm
How to verify uniqueness and existence for a linear elliptic PDE
Exercises
See chapter 21 in CDE.
Examination
1.1.
Show that the equation
−Δu = f,
x ∈ Ω
u = 0,
x ∈ Γ
satisfies the Lax­Milgram theorem (has a unique solution).
1.2.
The equation
−Δu = f,
x ∈ Ω
(∇u) ⋅ n = 0,
x ∈ Γ
does not satisfy the Lax­Milgram theorem (may not have a unique solution). Construct an agument showing
why not.
[TODO]
...
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This page was last modified on 23 September 2012, at 20:07.