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 17.75: Generating adaptively refined meshes: A posteriori error estimators

The ideal way to refine a mesh would be to look at the error on a cell and then refine those cells that have the largest error associated with them. But, there are numerous problems with this:

1. The error is defined as the difference between the exact and the computed finite element solution, u-uh, i.e., a function. To compare errors, we need to choose a norm of the error that gives us a number for each cell. How do we choose this?
2. A bigger problem is that we don't know the exact solution u, of course. If we did, we would not need the finite element method to compute an approximation. The question therefore becomes whether we can estimate a norm of the error using only computable quantities. This is the field of a posteriori error estimation.

This lecture discusses the second of these points, and also touches on the question whether refining a cell really decreases the overall error.