Approximation on Finite Elements Bruce A. Finlayson Rehnberg Professor of Chemical Engineering

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’

The function: looks like this -

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.

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

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.

Linear Finite Elements Improve the approximation by using a linear interpolation within the finite element.

Linear interpolations on 4, 8 and 16 finite elements

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).

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 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.)

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.