Lecture WS 21/22 Numerical Algorithms
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: hFEM 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, GramSchmidt, 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, GaussSeidel, 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
 FriedrichHirzebruchAllee 7, seminar room 2.035
Tutorials
 Date
 Thursdays, 16:15 to 17:45
 Location
 FriedrichHirzebruchAllee 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.