1. Weak Formulation ( variational formulation)
BVP
Weak Formulation ( variational formulation)
Multiply equation (1) by v(x) and then integrate from 0 to 1
Integration by parts yiels
where

2. Weak Formulation ( variational formulation)
Galerkin Methods
BVP
Weak Formulation ( variational formulation)
Unique sol?
Infinite dimensional space
where

3. Weak Formulation ( variational formulation)
Galerkin Methods
Weak Formulation ( variational formulation)
Infinite dimensional space
Is finite dim
Unique sol?

4. Galerkin Methods
Discrete Form
Is finite dim
We can approximate u

5. Galerkin Methods Linear system Linear system of equation square
Unique sol?
Linear system of equation
square
Symmetric (why)
Positive definite

6. Finite Element Methods1
0.25
0.5
0.75
why 1
0.25
0.5
0.75

7. 1 1 1 Finite Element Methods Finite Element Methods 1 1 1 0.25 0.51
0.25
0.5
0.75 1 1
0.25
0.5
0.75 1 1
0.25
0.5
0.75

8. Finite Element Methods
HomeWork:
Compute the matrix A and the vector b then solve the linear system and write the solution as a linear combination of the basis then approximate the value of the function at x= 0.3, 0.7. can you find the analytic solution of the problem?
where h=1/4 f(x)= x(x-1) with pc-linear