This is a semi-continuation of my previous blog post. My goal for Spring 2016 was to design and teach a numerical analysis course from the ground up. Lagrange interpolation with polynomials was to be the basis of this class.
After a semester, I would say this technique was a success. Polynomials proved to be a very simple way of deriving ODE and PDE methods. Because the class understood polynomial interpolation, differentiation and integration well, it was a breeze to use the Method of Weighted Residual framework to teach finite differences (FD), Chebyshev and Fourier spectral methods, radial basis function (RBF) spectral methods, and RBF-generated FD methods for PDEs.
At any point, to generate a numerical method, I could follow a consistent framework:
1. Pick a basis (polynomials, RBFs, Fourier).
2. Pick a set of collocation points (evenly-spaced, Chebyshev, scattered nodes).
3. Pick a support (global, compact) for that basis.
4. Pick a smoothness (piecewise-smooth, infinitely-smooth) for that basis.
5. Pick weight functions: delta functions for collocation methods, or the basis function itself for a Galerkin method.
Students reported enjoying the class and understanding the material well. Their HW performance reflected this as well.
The course notes for the class are located at http://www.math.utah.edu/~vshankar/5620.html.
Having taught the course for a semester, I now also have a basis (no pun intended) for the next time I teach it. I can think about adding more material on finite element and finite volume methods; I spent only one lecture on the former and no lectures on the latter this semester.