Bergische Universität Wuppertal
Fachbereich Mathematik und Naturwissenschaften
Angewandte Mathematik - Numerische Analysis (AMNA)

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Prof. Dr. M. Ehrhardt
D. Shcherbakov, M.Sc.

Lecture Course in Summer Term 2012:

Numerical Analysis and Simulation II: Partial Differential Equations (PDEs)



Outline
 
17.04.2012

§ 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
19.04.2012
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
24.04.2012

§ 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 Hk(Omega)
  • Definition 3: Sobolev-Space H0k(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
26.04.2012
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
03.05.2012
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:
08.05.2012
2.6 Stability and Convergence in the Maximum Norm
  • Theorem 1: Stability and Convergence in the Maximum Norm
10.05.2012

§ 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
15.05.2012
3.3 Dirichlet Problem in 2D
  • Definition 1: discretely connected
3.4 Discrete Maximum Principle
  • Lemma 1: Discrete Maximum Principle
  • Corollaries 1 to 6:
22.05.2012
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"
24.05.2012
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
5.06.2012

§ 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 Wpk(Omega)
  • Definition 1: Sobolev Spaces Wpk(Omega)
  • Definition 2: Lipschitz Boundary
  • Definition 3: Sobolev Spaces W0pk(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 Wq-k(Omega)
  • Lemma 1: Identification of Wq-k(Omega) with Dual Space of W0pk(Omega)
  • Definition 2: Sobolev-Slobodeckij-Space Hs(Omega)
4.5 The Theorem on equivalent Norms
  • Definition 1: equivalent Norms
  • Theorem 1: Theorem on equivalent Norms
  • Example 1: equivalent Norms in Wp1(Omega)
12.06.2012
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: Wpm(Omega) C Wpk(Omega), k<=m, 1<=p< infinity
  • Lemma 2: Wqk(Omega) C Wpk(Omega), k>=0, 1<=p<=q< infinity
  • Lemma 3: natural Extension
  • Theorem 1: Sobolev Inequality
  • Example 1:
  • Example 2:
14.06.2012

§ 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
19.06.2012
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
21.06.2012

§ 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
26.06.2012
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
28.06.2012

§ 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:
03.07.2012

§ 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 Vk
  • 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
05.07.2012
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
10.07.2012
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(nk)
12.07.2012
7.9 The CASCADE Algorithm


University of Wuppertal
Faculty of Mathematics and Natural Sciences
Department of Mathematics
Applied Mathematics & Numerical Analysis Group

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