This course provides a comprehensive introduction to the Finite Element Method (FEM) and modern solution techniques for both linear and nonlinear problems. It covers one- and two-dimensional field problems, as well as transient and solid mechanics applications. The module is designed to help students develop strong skills in FEM methodology, from fundamental concepts to practical computational implementation. Students are introduced to key principles such as strain energy, virtual work, and variational methods. Matrix analysis techniques and the Galerkin method are also presented as essential tools. The course explores linear structural elements, including springs, bars, trusses, and beams. Two-dimensional elements such as triangular and quadrilateral elements are studied in detail. Three-dimensional elements, including tetrahedral and solid elements, are also analyzed. The module further includes vibration analysis using FEM for different structural systems. Finally, advanced topics such as mesh design, convergence, material nonlinearity, and thermal problems are discussed.