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Lecture Mathematical Modelling and Simulation with Comsol Multiphysics II
AbstractContent of the lectureIntroduction into the theory of finite element methods
1. An application
1.1. Temperature distribution in a room
2. Mathematical notations and definitions
2.1. Some general notations
2.2. Spaces of continuous functions
2.3. Lebesgue spaces, embeddings and inequalities
2.4. Weak derivatives
2.5. Sobolev spaces, embeddings and inequalities
2.6. Divergence theorem and Green's formula
3. Basic Concepts of the finite element method
3.1. Weak formulation
3.2. Ritz-Galerkin method
3.3. Error estimates
3.3.1. Error estimates in energy norm
3.3.2. Error estimates in L2-norm
3.4. Space of continuous, piecewise linear polynomials
3.5. Implementation of Finite Element Methods
4. Variational formulation
4.1. General definitions and abstract linear spaces
4.2. Projection theorem
4.3. Riesz representation theorem
4.4. Abstract variational formulation
4.5. Lax-Milgram theorem
4.6. Céa's lemma
4.7. Appilcations to elliptic boundary value problems
4.7.1. Problems with Dirichlet boundary conditions
4.7.2. Problems with Neumann boundary conditions
4.7.3. Problems with Robin boundary conditions
5. Finite element spaces
5.1. Basic definition of a finite element
5.2. Triangular finite elements
5.2.1. The Lagrange element
5.2.2. The Hermite element
5.2.3. The Argyris element
5.3. Rectangular finite elements
5.3.1. The Lagrange element (the tensor product element)
5.3.2. The Serendipity element
5.4. The local interpolation operator
5.5. Triangulations and rectangular subdivisions
5.6. The finite element space
6. Convergence theory
6.1. The interpolation error
Content of the tutorialFreezing Relative Equilibria in Equivariant First-Order Evolution Equations (with Comsol Multiphysics 5.1 and 5.2)
1.1 Traveling waves in reaction diffusion systems
1.2 Freezing method for traveling waves
1.3 Numerical approximation of traveling waves via freezing method
1.4 Spectra and eigenfunctions of traveling waves
1.4.1 Point spectrum of traveling waves on the imaginary axis
1.4.2 Essential spectrum of traveling waves
2.1 Oscillating waves in reaction diffusion systems
2.2 Freezing method for oscillating waves
2.3 Numerical approximation of oscillating waves via freezing method
2.4 Spectra and eigenfunctions of oscillating waves
2.4.1 Point spectrum of oscillating waves on the imaginary axis
2.4.2 Essential spectrum of localized oscillating waves
3.1 Rotating waves in reaction diffusion systems
3.2 Freezing method for rotating waves
3.3 Numerical approximation of rotating waves via freezing method
4. Freezing Waves with Several Symmetries in Reaction Diffusion Systems
5. Freezing Multistructures (Multifronts, Multipulses and Multisolitons) and Wave Interactions in Reaction Diffusion Systems
6. Further coherent structures in Reaction Diffusion Systems
Freezing Relative Equilibria in Equivariant Second-Order Evolution Equations (with Comsol Multiphysics 5.1 and 5.2)
7.1 Traveling waves in systems of damped wave equations
7.2 Freezing method for traveling waves
7.3 Numerical approximation of traveling waves via freezing method
7.4 Spectra and eigenfunctions of traveling waves
7.4.1 Point spectrum of traveling waves on the imaginary axis
7.4.2 Essential spectrum of traveling waves
8. Freezing Rotating Waves in Damped Wave Equations
8.1 Rotating waves in systems of damped wave equations
8.2 Freezing method for rotating waves
8.3 Numerical approximation of rotating waves via freezing method
For the implementation we suggest the following tutorials Freezing Relative Equilibria in Equivariant First-Order Evolution Equations (with Comsol Multiphysics 5.1 and 5.2)1. Freezing Traveling Waves in Reaction Diffusion Equations
2. Freezing Oscillating Waves in Reaction Diffusion Equations
3. Freezing Rotating Waves in Reaction Diffusion Systems
4. Freezing Waves with Several Symmetries in Reaction Diffusion Systems
5. Freezing Multistructures (Multifronts, Multipulses and Multisolitons) and Wave Interactions in Reaction Diffusion Systems
6. Further coherent structures in Reaction Diffusion Systems
Freezing Relative Equilibria in Equivariant Second-Order Evolution Equations (with Comsol Multiphysics 5.1 and 5.2)7. Freezing Traveling Waves in Damped Wave Equations
8. Freezing Rotating Waves in Damped Wave Equations
LiteratureFinite element method (FEM):
Ich wünsche allen Teilnehmern der Vorlesung viel Erfolg! |
