- Prof. Dr. Willy Dörfler
- Prof. Dr. Marlis Hochbruck (responsible)
- Prof. Dr. Tobias Jahnke
- Prof. Dr. Wolfgang Reichel
- Prof. Dr. Christian Wieners
Time and Room
Thursday, 15:45-17:15 in Room 1C-04
The lecture on November 17 is moved to November 14 after the GRK seminar talk (16:30-17:15).
- Robust approximation of the Helmholtz equation
(Prof. Dr. Christian Wieners)
The 5 lectures (20.10, 27.10, 3.11., 10.11. and 14.11.) give an overview on recent results on the robust approximation of the Helmholtz equation. This is a classical problem: it is well known, that classical FE approximations are not robust in the high frequence case, the so-called 'pollution error' is dominant.
In the lecture, new methods to overcome this problem will be presented, summarizing the following two preprints:
On stability of discretizations of the Helmholtz equation (extended version),
Sofi Esterhazy, Jens Markus Melenk
Wavenumber Explicit Analysis for a DPG Method for the
Multidimensional Helmholtz Equation,
L. Demkowicz, J. Gopalakrishnan, I. Muga, and J. Zitelli
- On stability of discretizations of the Helmholtz equation (extended version),
- Bound states for a nonlinear curl-curl equation (Prof. Dr. Wolfgang Reichel)
In two lectures (24.11., 1.12. and 8.12.) I will explain the derivation of a nonlinear Schrödinger-type equation from Maxwell's equations in the presence of a material with nonlinear permittivity. The resulting equation contains the curl-curl operator and a nonlinearity of homogeneity-degree three. One then looks for stationary, exponentially localized (soliton-like) solutions via variational methods. Only few and partial results on existence of such solutions are available via exploiting symmetries of the equation. Most of the material of the lectures arise from the two papers:
- Azzollini, Antonio; Benci, Vieri; D'Aprile, Teresa; Fortunato, Donato Existence of static solutions of the semilinear Maxwell equations. Ric. Mat. 55 (2006), no. 2, 283–297.
- Benci, Vieri; Fortunato, Donato Towards a unified field theory for classical electrodynamics. Arch. Ration. Mech. Anal. 173 (2004), no. 3, 379–414.
- Splitting methods for time dependent pdes (Prof. Dr. Marlis Hochbruck)
We will discuss the numerical solution of the Gross-Pitaevskii equation. An error analysis for Strang splitting in time and Hermite collocation in space is presented. The 4 lectures will start on 15.12.
- Ludwig Gauckler, Convergence of a split-step Hermite method for the Gross-Pitaevskii equation, IMA J. Numer. Anal. 31 (2011), 396-415.
- Christian Lubich, On splitting methods for Schrödinger-Poisson and cubic nonlinear Schrödinger equations, Math. Comp. 77 (2008), 2141-2153.
- Ernst Hairer, Christian Lubich, and Gerhard Wanner, Geometric Numerical Integration, Springer Series in Computational Mathematics, Vol. 31, 2nd ed., 2006 (Calculus of Lie Derivatives in Section III.5)
- Sparse Grids (Prof. Dr. Tobias Jahnke)
The two lectors (26.1. and 2.2.2012) will be based on the following review paper:
- Hans-Joachim Bungartz and Michael Griebel, Sparse grids, Acta Numerica 13 (2004), 147-269.
- Commutative diagrams for Maxwell's equations (Prof. Dr. Willy Dörfler)