Contents -- 1 Introduction -- 1 Introduction to mathematical modelling and numerical simulation -- 1.1 General introduction -- 1.2 An example of modelling -- 1.3 Some classical models -- 1.3.1 The heat flow equation -- 1.3.2 The wave equation -- 1.3.3 The Laplacian -- 1.3.4 Schrödinger's equation -- 1.3.5 The Lamé system -- 1.3.6 The Stokes system -- 1.3.7 The plate equations -- 1.4 Numerical calculation by finite differences -- 1.4.1 Principles of the method -- 1.4.2 Numerical results for the heat flow equation -- 1.4.3 Numerical results for the advection equation -- 1.5 Remarks on mathematical models -- 1.5.1 The idea of a well-posed problem -- 1.5.2 Classification of PDEs -- 2 Finite difference method -- 2.1 Introduction -- 2.2 Finite differences for the heat equation -- 2.2.1 Various examples of schemes -- 2.2.2 Consistency and accuracy -- 2.2.3 Stability and Fourier analysis -- 2.2.4 Convergence of the schemes -- 2.2.5 Multilevel schemes -- 2.2.6 The multidimensional case -- 2.3 Other models -- 2.3.1 Advection equation -- 2.3.2 Wave equation -- 3 Variational formulation of elliptic problems -- 3.1 Generalities -- 3.1.1 Introduction -- 3.1.2 Classical formulation -- 3.1.3 The case of a space of one dimension -- 3.2 Variational approach -- 3.2.1 Green's formulas -- 3.2.2 Variational formulation -- 3.3 Lax-Milgram theory -- 3.3.1 Abstract framework -- 3.3.2 Application to the Laplacian -- 4 Sobolev spaces -- 4.1 Introduction and warning -- 4.2 Square integrable functions and weak differentiation -- 4.2.1 Some results from integration -- 4.2.2 Weak differentiation -- 4.3 Definition and principal properties -- 4.3.1 The space H[sup(1)](Ω) -- 4.3.2 The space H[sup(1)][sub(0)](Ω) -- 4.3.3 Traces and Green's formulas -- 4.3.4 A compactness result -- 4.3.5 The spaces H[sup(m)](Ω) -- 4.4 Some useful extra results.

4.4.1 Proof of the density theorem 4.3.5 -- 4.4.2 The space H(div) -- 4.4.3 The spaces W[sup(m,p)](Ω) -- 4.4.4 Duality -- 4.5 Link with distributions -- 5 Mathematical study of elliptic problems -- 5.1 Introduction -- 5.2 Study of the Laplacian -- 5.2.1 Dirichlet boundary conditions -- 5.2.2 Neumann boundary conditions -- 5.2.3 Variable coefficients -- 5.2.4 Qualitative properties -- 5.3 Solution of other models -- 5.3.1 System of linear elasticity -- 5.3.2 Stokes equations -- 6 Finite element method -- 6.1 Variational approximation -- 6.1.1 Introduction -- 6.1.2 General internal approximation -- 6.1.3 Galerkin method -- 6.1.4 Finite element method (general principles) -- 6.2 Finite elements in N = 1 dimension -- 6.2.1 P[sub(1)] finite elements -- 6.2.2 Convergence and error estimation -- 6.2.3 P[sub(2)] finite elements -- 6.2.4 Qualitative properties -- 6.2.5 Hermite finite elements -- 6.3 Finite elements in N ≥ 2 dimensions -- 6.3.1 Triangular finite elements -- 6.3.2 Convergence and error estimation -- 6.3.3 Rectangular finite elements -- 6.3.4 Finite elements for the Stokes problem -- 6.3.5 Visualization of the numerical results -- 7 Eigenvalue problems -- 7.1 Motivation and examples -- 7.1.1 Introduction -- 7.1.2 Solution of nonstationary problems -- 7.2 Spectral theory -- 7.2.1 Generalities -- 7.2.2 Spectral decomposition of a compact operator -- 7.3 Eigenvalues of an elliptic problem -- 7.3.1 Variational problem -- 7.3.2 Eigenvalues of the Laplacian -- 7.3.3 Other models -- 7.4 Numerical methods -- 7.4.1 Discretization by finite elements -- 7.4.2 Convergence and error estimates -- 8 Evolution problems -- 8.1 Motivation and examples -- 8.1.1 Introduction -- 8.1.2 Modelling and examples of parabolic equations -- 8.1.3 Modelling and examples of hyperbolic equations -- 8.2 Existence and uniqueness in the parabolic case.

8.2.1 Variational formulation -- 8.2.2 A general result -- 8.2.3 Applications -- 8.3 Existence and uniqueness in the hyperbolic case -- 8.3.1 Variational formulation -- 8.3.2 A general result -- 8.3.3 Applications -- 8.4 Qualitative properties in the parabolic case -- 8.4.1 Asymptotic behaviour -- 8.4.2 The maximum principle -- 8.4.3 Propagation at infinite velocity -- 8.4.4 Regularity and regularizing effect -- 8.4.5 Heat equation in the entire space -- 8.5 Qualitative properties in the hyperbolic case -- 8.5.1 Reversibility in time -- 8.5.2 Asymptotic behaviour and equipartition of energy -- 8.5.3 Finite velocity of propagation -- 8.6 Numerical methods in the parabolic case -- 8.6.1 Semidiscretization in space -- 8.6.2 Total discretization in space-time -- 8.7 Numerical methods in the hyperbolic case -- 8.7.1 Semidiscretization in space -- 8.7.2 Total discretization in space-time -- 9 Introduction to optimization -- 9.1 Motivation and examples -- 9.1.1 Introduction -- 9.1.2 Examples -- 9.1.3 Definitions and notation -- 9.1.4 Optimization in finite dimensions -- 9.2 Existence of a minimum in infinite dimensions -- 9.2.1 Examples of nonexistence -- 9.2.2 Convex analysis -- 9.2.3 Existence results -- 10 Optimality conditions and algorithms -- 10.1 Generalities -- 10.1.1 Introduction -- 10.1.2 Differentiability -- 10.2 Optimality conditions -- 10.2.1 Euler inequalities and convex constraints -- 10.2.2 Lagrange multipliers -- 10.3 Saddle point, Kuhn-Tucker theorem, duality -- 10.3.1 Saddle point -- 10.3.2 The Kuhn-Tucker theorem -- 10.3.3 Duality -- 10.4 Applications -- 10.4.1 Dual or complementary energy -- 10.4.2 Optimal command -- 10.4.3 Optimization of distributed systems -- 10.5 Numerical algorithms -- 10.5.1 Introduction -- 10.5.2 Gradient algorithms (case without constraints) -- 10.5.3 Gradient algorithms (case with constraints).

10.5.4 Newton's method -- 11 Methods of operational research (Written in collaboration with Stéphane Gaubert) -- 11.1 Introduction -- 11.2 Linear programming -- 11.2.1 Definitions and properties -- 11.2.2 The simplex algorithm -- 11.2.3 Interior point algorithms -- 11.2.4 Duality -- 11.3 Integer polyhedra -- 11.3.1 Extreme points of compact convex sets -- 11.3.2 Totally unimodular matrices -- 11.3.3 Flow problems -- 11.4 Dynamic programming -- 11.4.1 Bellman's optimality principle -- 11.4.2 Finite horizon problem -- 11.4.3 Minimum cost path, or optimal stopping, problem -- 11.5 Greedy algorithms -- 11.5.1 General points about greedy methods -- 11.5.2 Kruskal's algorithm for the minimum spanning tree problem -- 11.6 Separation and relaxation -- 11.6.1 Separation and evaluation (branch and bound) -- 11.6.2 Relaxation of combinatorial problems -- 12 Appendix: Review of hilbert spaces -- 13 Appendix: Matrix Numerical Analysis -- 13.1 Solution of linear systems -- 13.1.1 Review of matrix norms -- 13.1.2 Conditioning and stability -- 13.1.3 Direct methods -- 13.1.4 Iterative methods -- 13.1.5 The conjugate gradient method -- 13.2 Calculation of eigenvalues and eigenvectors -- 13.2.1 The power method -- 13.2.2 The Givens-Householder method -- 13.2.3 The Lanczos method -- Index -- A -- B -- C -- D -- E -- F -- G -- H -- I -- J -- K -- L -- M -- N -- O -- P -- Q -- R -- S -- T -- U -- V -- W -- Index notations.