Lecturer

Prof. Marc Alexander Schweitzer

Contact for exercises

Jannik Michels

Content

Learning targets

Broad overview and understanding of propositions, relations and methods from the area of numerical algorithms. Competence to evaluate the scope, utility, and limits of the methods and techniques and to independently apply abstract mathematical results to concrete problems. Competence to place the results in a more general mathematical context. Overview of connections to other areas and ability to arrive at rigorous mathematical proofs starting from heuristic considerations.

The selection of topics is based on the module handbook for the Master programme in mathematics.

Topics

• Repetition of classical finite element method (FEM) and functional analysis: h-FEM on regular meshes
• Fast solvers: domain decomposition, subspace correction, (geometric) multigrid, algebraic multigrid
• High order FEM and isogeometric analysis (IGA)
• Enriched approximations: partition of unity methods (PUM), extended FEM (XFEM), generalized FEM (GFEM)

Literature

• Dietrich Braess; Finite Elements: Theory, fast solvers, and applications in solid mechanics
• Dietrich Braess; Finite Elemente: Theorie, schnelle Löser und Anwendungen in der Elastizitätstheorie
• Wolfgang Hackbusch; Multigrid methods and applications
• JINCHAO XU; Iterative methods by Space Decomposition and Subspace correction (accessible in university network)
• Carl de Boor; B(asic)-Spline Basics

Prerequisites

There are no prerequisites for this lecture. However, topics and exercises of the preceding lectures Algorithmische Mathematik I (V1G5), Algorithmische Mathematik II (V1G6), V2E1 Einführung die Grundlagen der Numerik (V2E1) and Scientific Computing I (V3E1/F4E1) will be very helpful during this lecture.

These prerequisite topics include:

• Finite Element Method (FEM)

• Structured programming, in particular Python

• 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

• Preconditioning of iterative methods

• Eigenvalues, Eigenvectors, Bounds for the spectrum

Lecture times

Date

Tuesdays, 10:15 to 11:45

Thursdays, 8:15 to 9:45

Location

Friedrich-Hirzebruch-Allee 7, seminar room 2.035

Tutorials

Date

Thursdays, 16:15 to 17:45

Location

Friedrich-Hirzebruch-Allee 7, seminar room 2.035

Admittance for exam based on homework assignments requiring

• 50% of points from theory assignments.
• 50% of points from programming assignments.

eCampus

Any further information regarding the lecture can be found on eCampus.