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

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BVP

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  1. 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

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

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

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

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

  6. Finite Element Methods Finite Element Methods why 1 0 0.25 0.5 0.75 1 0 0.25 0.5 0.75

  7. 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

  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

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