Approximation on Finite Elements
The function x^2 exp(y-0.5) looks like this when plotted:
Approximation on finite elements
Here is what we expect in a contour plot of the function:
This is for N x N blocks, N=4
N =8
N = 16
N = 32
N = 64
N = 128
This is mesh refinement.
PPT Slide
Let functions in the block be bilinear functions of u and v.
N = 4, bilinear interpolation
N = 8, bilinear interpolation
N = 16, bilinear interpolation
Compare constant interpolation on finite elements with bilinear interpolation on finite elements.
Instead of matching the function at the block-corners, find the best interpolant minimizing the mean square difference between the approximation and the exact function. Still use finite elements, but bilinear approximations.
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.
Calculus of Variations
We choose finite element functions which satisfy the boundary conditions, and then find the values of the parameters that make the integral a minimum.
The solution with linear elements on 312 triangles (177 nodes) is:
The solution with linear elements on 1248 triangles (665 nodes) is:
Finite Element Variational Method
Galerkin Finite Element Method
Conclusion - Three Basic Ideas
Email: finlayson@cheme.washington.edu
Home Page: http://faculty.washington.edu/finlayso
Other information: Copyright, 2002, Bruce A. Finlayson
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