Elliptic and parabolic partial differential equations arise in a number of important physical problems, such as solid mechanics, fluid dynamics, heat flow, oil recovery and electrocardiology. Engineering texts for these methods tend to emphasize a specific application while describing the methods. Such a presentation can make it difficult to understand the general principles, and prevent students in other application areas from fully understanding the ideas. On the other hand, mathematics books on these methods tend to emphasize analysis, rather than applications. Treatment of boundary conditions, systems of partial differential equations, iterative methods and other important implementation details are largely ignored. In some cases, practical applications are ignored in mathematical methods texts.

This book grew out of lecture notes that I have been modifying for 35 years. The notes began as a fusion of lecture notes from my finite element mentors in mathematics, namely Jim Douglas Jr., Todd Dupont and Jim Bramble. The notes grew as I gained contact with engineers, particularly Eric Reissner and Gerry Frazier at UCSD, and Gary Trudeau at LLNL. I also benefitted from collaborations with mathematicians working on applications, such as Phil Colella and John Bell. I have written this book for others who are interested in applying numerical methods to physical problems, but want solid mathematical justification for their numerical schemes. Readers who are primarily interested in numerical methods for a particular application area, such as solid mechanics or fluid dynamics, will probably find other texts more suited to their needs.