This second edition has two parts. The first part is the complete classic by Gilbert Strang and George Fix, first published in 1973. The original book demonstrates the solid mathematical foundation of the finite element idea, and the reasons for its success. The second part is a new textbook by Strang. It provides examples, codes, and exercises to connect the theory of the Finite Element Method directly to the applications. The reader will learn how to assemble the stiffness matrix K and solve the finite element equations KU=F. Discontinuous Galerkin methods with a numerical flux function are now included. Strang's approach is direct and focuses on learning finite elements by using them.