BVP

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# BVP - PowerPoint PPT Presentation

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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## PowerPoint Slideshow about 'BVP' - clinton-cox

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Presentation Transcript

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

Galerkin Methods

Weak Formulation ( variational formulation)

Infinite dimensional space

Unique sol?

Is finite dim

Galerkin Methods

Discrete Form

Is finite dim

We can approximate u

Galerkin Methods

Linear system

Unique sol?

• Linear system of equation
• square
• Symmetric (why)
• Positive definite

Finite Element Methods

Finite Element Methods

why

1

0

0.25

0.5

0.75

1

0

0.25

0.5

0.75

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

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