Weak Formulation
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Weak Formulation. BVP. Weak Formulation ( variational formulation). Multiply equation (1) by v(x) and then integrate from 0 to 1. Integration by parts yiels. where. Galerkin Methods. BVP. Unique sol?. Weak Formulation ( variational formulation). Infinite dimensional space. where.

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Bvp

Weak Formulation

BVP

Weak Formulation ( variational formulation)

Multiply equation (1) by v(x) and then integrate from 0 to 1

Integration by parts yiels

where


Bvp

Galerkin Methods

BVP

Unique sol?

Weak Formulation ( variational formulation)

Infinite dimensional space

where


Bvp

Galerkin Methods

Weak Formulation ( variational formulation)

Infinite dimensional space

Unique sol?

Is finite dim


Bvp

Galerkin Methods

Discrete Form

Is finite dim

We can approximate u


Bvp

Galerkin Methods

Linear system

Unique sol?

  • Linear system of equation

  • square

  • Symmetric (why)

  • Positive definite


Bvp

Finite Element Methods

Finite Element Methods

why

1

0

0.25

0.5

0.75

1

0

0.25

0.5

0.75


Bvp

Finite Element Methods

Finite Element Methods

1

1

1

1

1

1

0

0

0

0.25

0.25

0.25

0.5

0.5

0.5

0.75

0.75

0.75


Bvp

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


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