NA from the Bottom-Up: Results

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

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.