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 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:

- The error is defined as the difference between the exact and the computed finite element solution,
*u-u*, i.e., a function. To compare errors, we need to choose a_{h}*norm*of the error that gives us a number for each cell. How do we choose this? - 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.

**Slides:** click here