- Repetition of classical finite element method (FEM) and functional analysis: h-FEM on regular meshes
- Fast solvers: Multigrid, Domain Decomposition
- High order FEM and isogeometric analysis (IGA): p-FEM, k-FEM
- Adaptive FEM: h-, hp-adaptive FEM
- Enriched Approximations: extended FEM (XFEM), generalized FEM (GFEM), partition of unity methods (PUM)
- Discontinuous Galerkin: Elliptic problems, conservation laws

- From smooth/regular problems and solutions
- to general/irregular/non-smooth/singular/discontinuous solutions

- Braess, Finite Elemente: Theorie, schnelle Löser und Anwendungen in der Elastizitätstheorie
- Braess, Finite Elements: Theory, fast solvers, and applications in solid mechanics
- Brenner, Scott: The mathematical theory of finite element methods
- Hackbusch: Elliptic Differential Equations: Theory and numerical treatment
- Hackbusch: Theorie und Numerik elliptischer Differentialgleichungen
- Hackbusch: Multigrid methods and applications
- Melenk: hp-finite element methods for singular perturbations
- Schwab: p- and hp-finite element methods: theory and applications in solid and fluid mechanics
- LeVeque: Numerical methods for conservation laws
- Xu: Iterative methods by Space Decomposition and Subspace correction (accessible in University network)
- Schweitzer: Meshfree and Generalized Finite Element Methods (Chapter 1, Section 2.2 for MLS)

Algorithmische Mathematik I (V1G5), Algorithmische Mathematik II (V1G6), V2E1 Einführung die Grundlagen der Numerik (V2E1)

These prerequisite topics include:

- Calculus, multidimensional differentiation and integration and Taylor expansion
- Elementary combinatorics and probability theory
- Structured programming, in particular C, Python
- Polynomial Interpolation
- Least Squares Approximation
- Conditioning of problems, Stability of algorithms
- Inner products, Orthogonality, Hilbert Spaces, Best approximation in Hilbert spaces
- Orthogonalization, Gram-Schmidt, Orthogonal Polynomials
- Numerical Integration, Quadrature
- Solution of systems of linear equations: Gauss elimination, LU decompostion, Cholesky decomposition, QR decomposition
- Solution of nonlinear equations: bisection, Newton
- Classical iterative methods for systems of linear equations:
**Jacobi**,**Gauss-Seidel**, Richardson, SOR - Krylov subspaces, Krylov subspace methods: Gradient descent,
**CG**,**PCG**, MINRES, GMRES - Preconditioning of iterative methods:
**PCG** - Eigenvalues, Eigenvectors, Bounds for the spectrum, Power iteration, Lanczos, Arnoldi, QR algorithm

Dates: | Tuesday | 10:15 – 11:45 |

Thursday | 08:30 – 10:00 | |

Location: | Wegelerstraße 6 - Seminarraum 6.020 | |

6th floor, last room on right, heading southwest |

Admittance for oral exam based on homework assignments requiring

- 50% of points from theory assignments
- 50% of points from programming assignments

Programming exercises will be based mainly in **Python**/**NumPy**/**matplotlib**.
Please send in your solutions to programming exercises via mail to your tutorial's teaching assistant.

This combination is a very useful for quick implementation. Algorithms can be put into code fast. Plots can be produced with little effort. Much of what is needed for the lecture can be found in the following examples.

Some Documentation and Tutorials can be found at the following links.- The Python Tutorial, Version 2.7.8
- NumPy Quickstart
- matplotlib tutorial

One easy way to obtain all necessary **Python** packages is Anaconda.

More installation alternatives, suggestions and instructions can be found on the websites for **NumPy** and **matplotlib**.

Model solutions are base on Python in version 2.7.8.
For editing and writing Python code, any good editor will do.
We recommend **Vim** or **Notepad++**.

- Graded oral exam, length 30 minutes in Besprechungszimmer, 6.007, Wegelerstraße 6
- Please do not forget to sign up for the exam via BASIS
- Registration for exam slots, first come first served, via lists at the end of the term
- successful participation in tutorials required for admittance to examination