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 3.98: The ideas behind the finite element method. Part 9: Sparsity as a key property

Discretizing partial differential equations via the finite element method leads to linear systems AU=F that can be very large: They can have millions or even billions of unknowns. A simple back-of-the-envelope calculation shows that we can not store matrices of this size, let alone solve linear systems with them.

How, then, do we deal with such problems? It turns out that the finite element method produces matrices that are sparse, that is, for which almost all of the entries of the matrix are zero. This lecture explains why this is so, and how that helps us store and solve even very large problems. In other words, sparsity is a key reason for the finite element method's success.