Elliptic differential equations : theory and numerical treatment / Wolfgang Hackbusch.

Preface; Contents; Chapter 1 Partial Differential Equations and Their Classification Into Types; 1.1 Examples; 1.2 Classification of Second-Order Equations into Types; 1.3 Type Classification for Systems of First Order; 1.4 Characteristic Properties of the Different Types; 1.5 Literature; Chapter 2 The Potential Equation; 2.1 Posing the Problem; 2.2 Singularity Function; 2.3 Mean-Value Property and Maximum Principle; 2.4 Continuous Dependence on the Boundary Data; Chapter 3 The Poisson Equation; 3.1 Posing the Problem; 3.2 Representation of the Solution by the Green Function

3.3 Existence of a Solution3.4 The Green Function for the Ball; 3.5 The Neumann Boundary-Value Problem; 3.6 The Integral Equation Method; Chapter 4 Difference Methods for the Poisson Equation; 4.1 Introduction: The One-Dimensional Case; 4.2 The Five-Point Formula; 4.3 M-matrices, Matrix Norms, Positive-Definite Matrices; 4.4 Properties of the Matrix Lh; 4.5 Convergence; 4.6 Discretisations of Higher Order; 4.7 The Discretisation of the Neumann Boundary-Value Problem; 4.7.1 One-Sided Difference for ∂u/∂n; 4.7.2 Symmetric Difference for ∂u/∂n; 4.7.3 Symmetric Difference for ∂u/∂n on an Offset

4.7.4 Proof of the Stability Theorem 4.624.8 Discretisation in an Arbitrary Domain; 4.8.1 Shortley-Weller Approximation; 4.8.2 Interpolation in Near-Boundary Points; Chapter 5 General Boundary-Value Problems; 5.1 Dirichlet Boundary-Value Problems for Linear Differential Equations; 5.1.1 Posing the Problem; 5.1.2 Maximum Principle; 5.1.3 Uniqueness of the Solution and Continuous Dependence; 5.1.4 Difference Methods for the General Differential Equation of Second Order; 5.1.5 Green's Function; 5.2 General Boundary Conditions; 5.2.1 Formulating the Boundary-Value Problem