This book is an introduction to the basic theory of the finite element method. In addition, it formulates a scheme for finite element computation of elasticity problems and provides typical computer programs for implementing the finite element method. The basic concepts and key techniques involved in finite element computation are presented to enable readers to quickly grasp the knowledge system of the finite element method. Main contents include: (1) introduction of finite element method; (2) fundamentals of elasticity mechanics; (3) weak form of equivalent integration; (4) elements and shape functions; (5) isoparametric element and numerical integration; (6) finite element computation scheme of elasticity problems; (7) solutions of linear algebraic equations; (8) error estimation and adaptive analysis; (9) programs of finite element method. This book is intended for undergraduate and postgraduate students majoring in engineering mechanics, civil engineering, and mining engineering. It provides also reference for researchers in related fields. To facilitate teaching and learning, this book provides selected homework exercises and programming projects with well-designed solutions. Online course and the source code of the related programs are also available.