Weak Formulation ( variational formulation)

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Presentation transcript:

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

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

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

Galerkin Methods Discrete Form Is finite dim We can approximate u

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

Finite Element Methods 1 0.25 0.5 0.75 why 1 0.25 0.5 0.75

1 1 1 Finite Element Methods Finite Element Methods 1 1 1 0.25 0.5 1 0.25 0.5 0.75 1 1 0.25 0.5 0.75 1 1 0.25 0.5 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