Approximation on Finite Elements

Outline - Finite Element Ideas

The function: looks like this -

Approximation on finite elements

Approximations with 4, 8, and 16 finite elements compared with the exact solution.

This is mesh refinement.

Linear Finite Elements

Linear interpolations on 4, 8 and 16 finite elements

To Review:

Instead of matching the function at the nodes, find the best interpolant minimizing the mean square difference between the approximation and the exact function.  Still use finite elements, but linear approximations within elements.

What do you do if you don’t know the function?  Suppose you want to minimize the difference between the approximation and exact function and their derivatives.

One can still find the best finite element approximation that minimizes this integral.  It won’t fit the function exactly anywhere, nor the first derivative, but it will minimize the integral.

Finite Element Variational Method

Conclusion - Three Basic Ideas