Copyright Page; Preface; Table of Contents; 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 differences2.3.1 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 discretization5.2 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 freedomA.2.2 Penalization technique; A.2.3 "Diagonalization" technique; A.2.4 Essential conditions in a vectorial problem; B Who's who; References; Subject Index