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 3.91: 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 fh,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 pth 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