Preconditioning and Partial Differential Equations

The purpose of this article is to explain how some apparently simple problems from numerical linear algebra are in fact extremely difficult, so that we cannot hope to solve them effectively in general. However, if we build and analyze algorithms to solve them in special cases of interest for the numerical analysis of partial differential equations, we find that the theory needed to validate these methods is of the same nature as that used for pseudo-differential operators, though the operators considered probably do not enter the framework of pseudo-differential operators. This relationship between distant fields of analysis displays how the interaction between different parts of mathematics, motivated by problems of practical origin, leads to interesting questions and solutions. The article is intended for mathematicians of all backgrounds and is written so that beginning graduate students with a good background in PDE’s and analysis can read it, and hopefully enjoy it also.