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 35: What preconditioner to use — Introduction and Parts 1+2: Simple preconditioners for simple problems
The performance of iterative solvers depends crucially on properties of the matrix of the linear system to be solved. In order to align these properties with the cases where iterative solvers work well, we typically "precondition" linear systems by multiplying it through with a matrix intended to approximate A-1.
In this lecture, I present the general idea of preconditioners and how one can come up with preconditioners — based on centuries-old defect correction methods or incomplete decompositions of the matrix — for simple problems such as the Laplace equation. In the second part, I also discuss other, scalar or elliptic problems.
Slides: click here