Part I Basic Material. 1 Some fundamental tools. 1.1 Hilbert spaces. 1.2 Distributions. 1.3 The spaces Lp and Hs. 1.4 Sequences in lp. 1.5 Important inequalities. 1.6 Brief overview of matrix algebra. 2 Fundamentals of finite elements and finite differences. 2.1 The one dimensional case: approximation by piecewise polynomials. 2.2 Interpolation in higher dimension using finite elements. 2.2.1 Geometric preliminary definitions. 2.2.2 The finite element. 2.2.3 Parametric Finite Elements. 2.2.4 Function approximation by finite elements. 2.3 The method of finite differences. 2.3.1 Difference quotients in dimension one. Part II Stationary Problems. 3 Galerkin-finite element method for elliptic problems. 3.1 Approximation of 1D elliptic problems. 3.1.1 Finite differences in 1D. 3.2 Elliptic problems in 2D. 3.3 Domain decomposition methods for 1D elliptic problems. 3.3.1 Overlapping methods. 3.3.2 Non-overlapping methods. 4 Advection-diffusion-reaction (ADR) problems . 4.1 Preliminary problems. 4.2 Advection dominated problems. 4.3 Reaction dominated problems. Part III Time dependent problems. 5 Equations of parabolic type. 5.1 Finite difference time discretization. 5.2 Finite element time discretization. 6 Equations of hyperbolic type. 6.1 Scalar advection-reaction problems. 6.2 Systems of linear hyperbolic equations of order one. 7 Navier-Stokes equations for incompressible fluids. 7.1 Steady problems. 7.2 Unsteady problems. Part IV Appendices. A The treatment of sparse matrices. A.1 Storing techniques for sparse matrices. A.1.1 The COO format. A.1.2 The skyline format. A.1.3 The CSR format. A.1.4 The CSC format. A.1.5 The MSR format. A.2 Imposing essential boundary conditions. A.2.1 Elimination of essential degrees of freedom. A.2.2 Penalization technique. A.2.3 “Diagonalization” technique. A.2.4 Essential conditions in a vectorial problem. B Who’swho.