Approximation on Finite Elements

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Approximation on Finite Elements

• Bruce A. Finlayson
• Rehnberg Professor of
• Chemical Engineering

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Outline - Finite Element Ideas

• Interpolation - piecewise constant and linear
  functions
• Mesh refinement
• Either fit the function at the nodes or minimize
  an integral to find the best fit

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The functionlooks like this -
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Approximation on finite elements

• Break the line 0 x 1 into small regions and
  approximate the function as a constant in that
  region, called a finite element.
• The approximation depends on the number of finite
  elements.

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Approximations with 4, 8, and 16 finite elements
compared with the exact solution.
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This is mesh refinement.

• Notice how the picture got better and better the
  more small regions we took.
• We approximated the function on each region - a
  finite element approximation.
• We get a better approximation when we use small
  finite elements.
• As the number of finite elements increases, the
  picture approaches that of a continuous function.

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Linear Finite Elements

• Improve the approximation by using a linear
  interpolation within the finite element.

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Linear interpolations on 4, 8 and 16 finite
elements
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To Review
We can improve the approximation by using more,
and smaller finite elements, (16 elements)
or by increasing the degree of polynomial in the
element (8 elements).
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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.
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What do you do if you dont know the function?
Suppose you want to minimize the difference
between the approximation and exact function and
their derivatives.
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One can still find the best finite element
approximation that minimizes this integral. It
wont fit the function exactly anywhere, nor the
first derivative, but it will minimize the
integral.
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Finite Element Variational Method

• Divide the domain into small regions.
• Write a low degree polynomial on each small
  region constant, linear, quadratic. These are
  the basis functions.
• Write the solution as a series of basis
  functions.
• Determine the coefficients by minimizing an
  integral. (The trick is to know what integral to
  use.)

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Conclusion - Three Basic Ideas

• Write the solution in a series of functions, each
  of which is defined over small elements, using
  low-order polynomials.
• Fit the functions to desired values at nodes or
  minimize an integral.
• Increase the number of basis functions in order
  to show convergence.