Basic Concepts and Methods for PDEs’ Approximation.- Numerical Solution of Linear Systems.- Finite Element Approximation.- Polynomial Approximation.- Galerkin, Collocation and Other Methods.- Approximation of Boundary Value Problems.- Elliptic Problems: Approximation by Galerkin and Collocation Methods.- Elliptic Problems: Approximation by Mixed and Hybrid Methods.- Steady Advection-Diffusion Problems.- The Stokes Problem.- The Steady Navier-Stokes Problem.- Approximation of Initial-Boundary Value Problems.- Parabolic Problems.- Unsteady Advection-Diffusion Problems.- The Unsteady Navier-Stokes Problem.- Hyperbolic Problems.