1. DE Weak Form Linear System
Two-Point Boundary Value Problem DE
Weak Form 1 2
Linear System
Discrete Form 4 3

2. spaces Definition: Definition: Remark:
The space of all square integrable funcions defined in the domain
Definition:
The function and its first derivative are square integrable
Remark:
Both spaces are Hilbert spaces. R2 is also a Hilbert space
R2 is also a Hilbert space
with inner product

3. Triangle inequality Triangle inequality: Triangle inequality:

4. Cauchy-Schwarz inequality
Cauchy-Schwarz inequality: (integral form)
Example: verify CS-inequality

5. Cauchy-Schwarz inequality
Is this true?:
Cauchy-Schwarz inequality: (integral form)

6. Bilinear form Definition: Definition: Definition: Example: Example:
The bilinear form is said to be symmetric if
a(w, v) = a(v,w), ∀v,w ∈ V,
A bilinear form on V is a function : V × V → R, which is linear in each argument separately
Definition:
the bilinear form a(・, ・) on V is bounded if there is a constant M such that.
Example:
Example:
prove that a is bounded bilinear form on

7. Bilinear form Linear functional Definition: Definition: Example:
the bilinear form a(・, ・) on is coercive if there is a constant α > 0 such that.
A linear functional L : V → R is said to be bounded
is the smallest constant c
Example:
prove that a is coercive on
Remark:
Example:

8. Lax-Milgram lemma 2 Lax-Milgram lemma Consider: where Hilbert space
bilinear form on
linear functional on
Lax-Milgram lemma
Hilbert space
bounded coercive bilinear form on
bounded linear functional on
Then
there exists a unique vector u ∈ V such that (2) is satisfied

9. Lax-Milgram lemma 1 2 2 1 DE Weak Form Example:
Show that there exist a unique solution for (2)

10. Lax-Milgram lemma 2 1 Example: solution:
Show that there exist a unique solution for (2)
In order to show that there exist a unique solution for (2), we need to satisfy all the conditions of Lax-Milgram lemma
solution:
Poincare’s inequality (HW)
proof
Show that:
Later we will do another proof for a symmetric a

12. Lax-Milgram lemma 2 1 3 Example:
Show that there exist a unique solution for (3) 3
Thm: A finite dimensional subspace of a Hilbert space is Hilbert
solution:
In order to show that there exist a unique solution for (3), we need to satisfy all the conditions of Lax-Milgram lemma

13. Stability 2 Example: Definintion: Stability:
A problem that satisfies the three conditions is said to be well posed
1)existence of solutions,
2)uniqueness of solutions,
3)stability 2
Poincare’s inequality
Stability:
continuous dependence of solutions with respect to perturbations of data
Solution bounded by the data of the problem
Setting ϕ = u in (2) and using
(coercive) and (Poincare), we find
Small change in the data produce small chang in the solution

14. Linear System of Equations4 3
Remark:
A is symmetric and positive definite
Remark:
Definition:
Under what condition that
(4) has solution
An nxn matrix A is symmetric and positive definite if
(4) has a unique solution iff that the matrix A is invertible ( non-singular )
Example: show that A is SPD

15. Linear System of Equations4 3
Remark:
A is symmetric and positive definite
Remark:
Proof:
Under what condition that
(4) has solution
(4) has a unique solution iff that the matrix A is invertible ( non-singular )
coercively