This book is devoted to the finite element method (FEM), an important direction in the numerical solution of partial differential equations (PDE). The author combines theoretical foundations and practical algorithms, explaining the essence of the method step by step. The work describes: General principles of solving elliptic, parabolic and hyperbolic PDEs; Variational formulas and their mathematical foundations; Discretization, i.e., the representation of continuous problems on a finite element grid; Stability and convergence properties; Practical computational algorithms and their application on computers; Methods of application to engineering and physics problems.