Starting in the spring 2013, I videotaped the lectures
for my *MATH 676: Finite element methods in scientific
computing* course
at the KAMU TV studio at
Texas A&M. These are lectures
on many aspects of scientific computing, software,
and the practical aspects of the finite element method, as
well as their implementation in the
deal.II software library. Support
for creating these videos was also provided by the
National Science Foundation and
the Computational
Infrastructure in Geodynamics.

**Note 1:** In some of the videos, I demonstrate code or user
interfaces. If you can't read the text, change the
video quality by clicking on the "gear" symbol at the
bottom right of the YouTube player.

**Note 2:**
deal.II is an
actively developed library, and in the course of this
development we occasionally deprecate and remove
functionality. In some cases, this implies that we also
change tutorial programs, but the nature of videos is that
this is not reflected in something that may have been
recorded years ago. If in doubt, consult
the *current* version of the tutorial.

**Lecture 3.9: The ideas behind the finite element method. Part 1: Approximation**

While these lectures are not meant as a complete course on the finite element method, it is worth discussing some of the underlying principles and ideas of the finite element method. Specifically, there are two key ideas to it:

- We seek a
*function*that approximates the exact solution. How do we approximate best? It turns out that piecewise polynomial approximation is the way to go. - Once we know what
*kind*of approximation we want, we need to say how we are actually going to*find*this approximate solution.

This lecture is about the first of these two points: Approximating the solution, and specifically why piecewise polynomial approximation is such a good idea.

**Slides:** click here