§ 1. Introduction

1.1 Definitions, Examples

• Definition 1: Partial Differential Equation
• Example 1: Partial Differential Equation of first Order
• Example 2: Laplace-Gleichung
• Example 3: Wave Equation
• Example 4: Heat Equation
• Example 5: Stokes Equation
• Example 6: Euler Equation

1.2 Type Classification for Second Order Equations

• Definition 1: elliptic, hyperbolic, parabolic
• Example 1: Laplace-/Wave-/Heat Equation
• Definition 2: Type Classification for n independent Variables

1.3 Type Classification for First Order Systems

• Definition 1: real-diagonalizable
• Definition 2: elliptic, hyperbolic
• Example 1: Cauchy-Riemann-Equations
• Example 2: Heat Equation

§ 2. Difference Methods for Parabolic Differential Equations

2.1 Mathematical Models

• Example 1: Heat Transport
• Example 2: Diffusion

2.2 Classical and Weak Solutions

• Definition 1: classical Solution
• Example 1: Heat Equation
• Definition 2: Sobolev-Space H^k(Omega)
• Definition 3: Sobolev-Space H₀^k(Omega)
• Example 2: Heat Equation
• Definition 4: Weak Solution
• Definition 5: continuous Bilinear Form
• Definition 6: V-elliptic Bilinear Form
• Theorem 1: Existence and Uniqueness of Weak Solutions

2.3 Difference Methods for onedimensional Parabolic Problems

• Definition 1: Grid, grid function
• Definition 2: Difference quotient
• Definition 3: (order of) consistency of a difference operator
• Example 1: Lu = d/dx (k(x) du/dx)
• Lemma 1: order of consistency of theta-scheme for heat equation

2.4 Stability and Convergence in l²

• Definition 1: Stability of a Method
• Example 1: Stability Analysis following von Neumann (discrete separation of variables)
• Example 2: formal Fourier-Stability technique following von Neumann
• Lemma 1: l²-Stability of the θ-Scheme (w.r.t. the Initial Condition)
• Lemma 2: l²-Stability of the θ-Scheme (w.r.t. the right hand side and the Initial Condition)
• Theorem 1: Consistency and Stability jointly leads to Convergence

2.5 Tridiagonal Systems

• Lemma 1: Sufficient Conditions for Feasibility and Stability
• Lemma 2: Discrete Maximum principle
• Corollaries 1 to 4:

2.6 Stability and Convergence in the Maximum Norm

• Theorem 1: Stability and Convergence in the Maximum Norm

§ 3. Difference Methods for elliptic Differential Equations

3.1 Mathematical Models

• Example 1: Heat Transport or Diffusion
• Example 2: Vibrations
• Example 3: Diffraction
• Example 4: Displacement of a Membrane

3.2 Difference Approximation of the Laplace Operators

• Approximation at curvilinear Boundaries

3.3 Dirichlet Problem in 2D

• Definition 1: discretely connected

3.4 Discrete Maximum Principle

• Lemma 1: Discrete Maximum Principle
• Corollaries 1 to 6:

3.5 Stability and Convergence

• Lemma 1: Maximum Norm-Stability (w.r.t. the right hand side and the Initial Condition)

3.6 Dirichlet Problem in a Rectangle

• Solution of the Eigenvalue Problem, Condition in the Spectral Norm
• Solution Method based on the "Fast Fourier Transform"

3.7 Discretizations of higher Order, compact Scheme

• Compact Scheme of higher Order
• Lemma 1: Embedding theorem
• Theorem 1: Convergence of the Scheme of higher Order

§ 4. Introduction to the Theory of Sobolev-Spaces

4.1 The generalized Derivative

• Definition 1: Space of Test Functions
• Definition 2: Convergence in the Space of Test Functions
• Definition 3: generalized Derivative
• Example 1: Derivative of the Hat Function
• Definition 4: Distributions
• Example 2: regular / singular Distribution
• Example 3: Dirac Delta Function
• Definition 5: distributive Derivative

4.2 The Sobolev Spaces W_p^k(Omega)

• Definition 1: Sobolev Spaces W_p^k(Omega)
• Definition 2: Lipschitz Boundary
• Definition 3: Sobolev Spaces W0_p^k(Omega)

4.3 The generalized Boundary Function

• Lemma 1: Trace Mapping

4.4 Sobolev Spaces with non-integer and negative Order

• Definition 1: Sobolev Spaces W_q^-k(Omega)
• Lemma 1: Identification of W_q^-k(Omega) with Dual Space of W0_p^k(Omega)
• Definition 2: Sobolev-Slobodeckij-Space H^s(Omega)

4.5 The Theorem on equivalent Norms

• Definition 1: equivalent Norms
• Theorem 1: Theorem on equivalent Norms
• Example 1: equivalent Norms in W_p¹(Omega)

4.6 Some Inequalities in Sobolev Spaces

• Lemma 1: (generalized) Friedrichs Inequality
• Lemma 2: Poincaré Inequality

4.7 The Integration by Parts Formula

• Lemma 1: Balance Identity
• Corollaries 1 to 4: Integration by Parts, Greens' Formula

4.8 Embedding Theorems and Sobolev Inequality

• Definition 1: continuously embedded
• Lemma 1: W_p^m(Omega) C W_p^k(Omega), k<=m, 1<=p< infinity
• Lemma 2: W_q^k(Omega) C W_p^k(Omega), k>=0, 1<=p<=q< infinity
• Lemma 3: natural Extension
• Theorem 1: Sobolev Inequality
• Example 1:
• Example 2:

§ 5. Variational Formulation of Boundary Value Problems

5.1 Bilinear Forms

• Definition 1: Bilinear Form
• Example 1: Vector Space with Scalar products
• Definition 2: Hilbert Space
• Definition 3: continuous (bounded) Bilinear Form
• Definition 4: coercive Bilinear Form
• Lemma 1: Subspace with coercive Bilinear Form is Hilbert Space

5.2 Projection on Subspaces

• Theorem 1: Theorem on Projections
• Definition 1: Projector

5.3 Representation Theorem of Riesz

• Theorem 1: Representation Theorem of Riesz

5.4 Variational Formulations

• Theorem 1: Existence and Uniqueness of Solutions to the Variational Problems
• Theorem 2: Existence and Uniqueness of Solutions to the Galerkin Method
• Corollary 1: Fundamental Orthogonality
• Corollary 2: Infimum in Energy Norm is attained
• Corollary 3: Ritz Approach
• Example 1: Boundary Value Problem in 1D, non-symmetric Bilinear Form
• Theorem 3: Lax-Milgram
• Corollary: Galerkin Equation
• Theorem 4: Lemma of Céa

5.5 Examples of elliptic Variational Formulations

• Example 1: Dirichlet Problem for the Poisson Equation
• Example 2: Neumann Problem

§ 6. The Finite Element Method

6.1 Basics, Definitions

• Definition 1: finite Element, Element Domain, Form Functions, Knot variabls
• Example 1: Lagrange Element in 1D
• Definition 2: knot basis
• Lemma 1: Charakterization of a Basis of P'

6.2 Triangle Elements in R²

• Example 1: Lagrange Triangle
• Example 2: quadratic Lagrange Triangle
• Example 3: cubic Hermite Element

6.3 Tetraeder Elements in R³

• Example 1: Lagrange Tetraeder
• Example 2: quadratic Lagrange Tetraeder

6.4 Triangulation for Subsets of R²

• Definition 1: feasible Triangulation
• Example 1: unfeasible Triangulation
• Lemma 1:

6.5 Convergence Conformal finite Elements

• Definition 1: affine-equivalent Sets
• Lemma 1: Change of Sobolev Seminorms by affine Mapping (Transformation Theorem )
• Example 1:
• Lemma 2: Geometric Estimates of the Norms ||B|| and ||B^-1||
• Example 1 (Continuation):
• Theorem 1: Convergence of the Finite Element Method

6.6 L²-Estimate for linear Elements

• Theorem 1: Convergence of linear Elements in L²

§ Discontinuous Galerkin Method (DG-Method)

A Model Problem

• Variational Formulation
• Galerkin Approximation and Galerkin-Orthogonality
• Example 1:
• Example 2:
• Example 3:
• Error Estimates and Step Size Control
• Definition 1:

§ 7. Introduction to Multigrid Methods

7.1 A Model Problem

• Idea of the Multigrid Methods

7.2 Grid dependent Norms

• Definition 1: Grid dependent Skalar product (.,.)_k
• Lemma 1: by (.,.)_k induced Norm is equivalent to the L²-Norm in V_k
• Lemma 2: generalized Cauchy-Schwarz Inequality

7.3 The Method of Simple Iteration

7.4 The Multigrid Algorithm

• Definition 1: Interpolating Operator, Projector
• Definition 2: Orthoprojector
• Lemma 1: Properties of the Orthoprojector
• Lemma 2: Properties of the Remainderprojector
• Lemma 3: Estimate of the Remainderprojector

7.5 The Convergence of the W-Cycle

• Lemma 1: Estimate for the relaxation operator
• Corollary: Convergence of the TwoGrid (TG) method in the energetic norm
• Theorem 1: Convergence of the multigrid method

7.6 The Convergence of the W-Cycle

• Lemma 1: Equation for the Error Operator
• Lemma 2: Relaxationoperator is self adjoint in (.,.)_E=a(.,.)
• Lemma 3: Error operator is self adjoint in (.,.)_E and positive semi-definite
• Lemma 4: Eigenvalues of the error operator
• Theorem 1: Konvergence dof the V-cycle

7.7 The Convergence of the "Full-Multigrid"-Method

• Theorem 1: Convergence of the "Full-Multigrid"-Method

7.8 The Computational Effort

• Theorem 1: Number of arithmetic Operations is Op(k)=O(n_k)

7.9 The CASCADE Algorithm