This book gives an introduction to the finite element method as a general method for the numerical solution of partial differential equations in science and engineering. The finite element method has been developed over the last twenty years and is today a dominating technique in computational mathematics with important applications in many areas. The book gives the mathematical foundation of the method together with important numerical aspects but there is also a clear connection with applications and many examples from various fields are considered. A strong effort has been made to make the presentation of the mathematics as non-technical and easily accessible as possible, while still maintaining a mathematical framework stressing fundamenta! concepts as stability, regularity etc. All the basic linear partial differential equations are considered, i e, elliptic, parabolic and hyperbolic type problems, stationary as well as time-dependent problems. There is also one chapter on finite element methods for integral equations (boundary element methods) and an introduction to nonlinear problems The book contains unique recent developments of finite element techniques to parabolic problems including methods for automatic time step control, and in particular to hyperbolic problems with important applications to e g fluid mechanics.