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.

The videos are part of a broader effort to develop a modern way of teaching Computational Science and Engineering (CS&E) courses. If you are interested in adapting our approach, you may be interested in this paper I wrote with a number of education researchers about the structure of such courses and how they work.

**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 3.91: The ideas behind the finite element method. Part 2: Theory of (piecewise) polynomial approximation**

The previous lecture talked about why piecewise polynomial approximation is one of the two key principles that underlies the finite element method. This lecture is now about a bit of theory on polynomial approximation. Specifically, if we are given a function *f(x)*, how accurate is a polynomial approximation *f _{h,p}(x)* of polynomial degree

*p*on a mesh with maximal cell diameter

*h*?

**Note:** In the slides shown during the lecture, I accidentally show that in the right hand side of the interpolation error estimate, one needs the *p*th derivative *f ^{(p)}*. That is wrong: It should have been the

*(p+1)th derivative. The slides linked to below correct this mistake.*

**Slides:** click here