Theoretical Numerical Analysis

1 Linear Spaces.- 1.1 Linear spaces.- 1.2 Normed spaces.- 1.3 Inner product spaces.- 1.4 Spaces of continuously differentiable functions.- 1.5 Lp spaces.- 1.6 Compact sets.- 2 Linear Operators on Normed Spaces.- 2.1 Operators.- 2.2 Continuous linear operators.- 2.3 The geometric series theorem and its variants.- 2.4 Some more results on linear operators.- 2.5 Linear functional.- 2.6 Adjoint operators.- 2.7 Types of convergence.- 2.8 Compact linear operators.- 2.9 The resolvent operator.- 3 Approximation Theory.- 3.1 Interpolation theory.- 3.2 Best approximation.- 3.3 Best approximations in inner product spaces.- 3.4 Orthogonal polynomials.- 3.5 Projection operators.- 3.6 Uniform error bounds.- 4 Nonlinear Equations and Their Solution by Iteration.- 4.1 The Banach fixed-point theorem.- 4.2 Applications to iterative methods.- 4.3 Differential calculus for nonlinear operators.- 4.4 Newton's method.- 4.5 Completely continuous vector fields.- 4.6 Conjugate gradient iteration.- 5 Finite Difference Method.- 5.1 Finite difference approximations.- 5.2 Lax equivalence theorem.- 5.3 More on convergence.- 6 Sobolev Spaces.- 6.1 Weak derivatives.- 6.2 Sobolev spaces.- 6.3 Properties.- 6.4 Characterization of Sobolev spaces via the Fourier transform.- 6.5 Periodic Sobolev spaces.- 6.6 Integration by parts formulas.- 7 Variational Formulations of Elliptic Boundary Value Problems.- 7.1 A model boundary value problem.- 7.2 Some general results on existence and uniqueness.- 7.3 The Lax-Milgram lemma.- 7.4 Weak formulations of linear elliptic boundary value problems.- 7.5 A boundary value problem of linearized elasticity.- 7.6 Mixed and dual formulations.- 7.7 Generalized Lax-Milgram lemma.- 7.8 A nonlinear problem.- 8 The Galerkin Method and Its Variants.- 8.1 The Galerkin method.- 8.2The Petrov-Galerkin method.- 8.3 Generalized Galerkin method.- 9 Finite Element Analysis.- 9.1 One-dimensional examples.- 9.2 Basics of the finite element method.- 9.3 Error estimates of finite element interpolations.- 9.4 Convergence and error estimates.- 10 Elliptic Variational Inequalities and Their Numerical Approximations.- 10.1 Introductory examples.- 10.2 Elliptic variational inequalities of the first kind.- 10.3 Approximation of EVIs of the first kind.- 10.4 Elliptic variational inequalities of the second kind.- 10.5 Approximation of EVIs of the second kind.- 11 Numerical Solution of Fredholm Integral Equations of the Second Kind.- 11.1 Projection methods: General theory.- 11.2 Examples.- 11.3 Iterated projection methods.- 11.4 The Nyström method.- 11.5 Product integration.- 11.6 Projection methods for nonlinear equations.- 12 Boundary Integral Equations.- 12.1 Boundary integral equations.- 12.2 Boundary integral equations of the second kind.- 12.3 A boundary integral equation of the first kind.- References.