Thefinite element method is one of the major tools used in the numerical solution of partial differential equations. This book offers a fundamental and practical introduction to the method, its variants, and their applications. In presenting the material, I have attempted to introduce every concept in the simplest possible setting and to maintain a level of treatment that is as rigorous as possible without being unnecessarily abstract. The book is based on the material that I have used in a graduate course at Southern Methodist University for several years. Part of the material was alsousedformyseminarnotesatPurdueUniversity,UniversityofMinnesota, andTexasA&MUniversity.Furthermore,thisbookwasthebasisforsummer schools on the finite element method and its applications held in China, Iran, Mexico, and Venezuela. This book covers six major topics and four applications. In Chap. 1, the 1 2 standard (H-and H -conforming)finite element method is introduced. In Chaps.2and3,twocloselyrelatedfiniteelementmethods,thenonconforming and the mixedfinite element methods, are discussed. The discontinuous and characteristicfiniteelementmethodsarestudiedinChaps.4and5;thesetwo methods have been recently developed. The adaptivefinite element method is considered in Chap. 6. The last four chapters are devoted to applications of these methods to solid mechanics (Chap. 7),—fluid mechanics (Chap. 8),—fluid flow in porous media (Chap. 9), and semiconductor modeling (Chap. 10).