These are some of the topics of our interests and current work.
Don't hesitate to contact us if you need help or are interested in a cooperation
in one of these or related topics.

The mathematical model is given by the van Roosbroeck equations
\begin{align*}
-\nabla \cdot \varepsilon \nabla w
&= C - n - p
,\\
\frac{\partial n}{\partial t} - \nabla \cdot n_i \mu_n e^w \nabla e^{-\phi_n}
&= R
,\\
\frac{\partial p}{\partial t} - \nabla \cdot n_i \mu_p e^{-w} \nabla e^{\phi_p}
&= R,
\end{align*}
where \(w\) is the electrostatic potential, \(n = n_i e^{w - \phi_n}\) and \(p = n_i e^{\phi_p - w}\)
are the electron and hole densities, \(C\) is doping, \(\phi_n\) and \(\phi_p\) are quasi-Fermi potentials,
and \(R = r(\boldsymbol{x},n,p) \left( n_i^2 - np \right)\) is the recombination/generation rate with
\(r(\boldsymbol{x},n,p) \geq 0\).

For the numerical simulation, it is essential for the discrete
system to preserve essential properties of the analytical system,
such as the maximum principle, positivity of the carrier
concentrations, and current conservation.

Oskar: semiconductor device simulator

The core idea behind Oskar is the consideration
of the physical structure in the numerical solution of the underlying
semiconductor equations.
Similar structures can be also found in chemical systems
and in the image processing (see a phase separation example below).
The Voronoi FVM allows to carry over the essential properties
of the analytical system to the discretized equations.

Oskar's strength has contributed significantly to the worldwide success of the
MPG HLL
detectors, including projects for
the
SLAC National Accelerator Laboratory (example),
the European X-ray Free Electron Laser research facility
XFEL
(example),
the High-Energy Accelerator Research Organization
KEK,
and the Advanced Telescope for High-Energy Astrophysics
Athena
of the European Space Agency (example).

Oskar was also successfully used in cooperation with the
Institute of Applied Photophysics
at the TU Dresden to gain a deeper understanding of the mode of operation and major
limitations of doped organic semiconductors such as organic light emitting diodes (OLEDs),
permeable base transistors (PBTs) and vertical organic field effect transistors (VOFETs).

The mathematical description for the non-local phase segregation problem is given by
\begin{align}
\begin{cases}
\frac{\partial u}{\partial t}
- \nabla \cdot \left(
f (\lvert \nabla v \rvert)
\left( \nabla u + \frac {\nabla w}{\phi''(u)} \right) \right) = 0
~\text{ in $\Omega$}
,
\\
u(0,\cdot) = u_0(\cdot)
,
\\
v = \phi'(u) + w
,
\\
w(t,\boldsymbol{x}) = \int_\Omega \mathcal{K} \left( \lvert \boldsymbol{x}-\boldsymbol{y} \rvert \right)
\left(1 - 2 u(t,\boldsymbol{y}) \right) \, \mathrm{d} \boldsymbol{y}
,
\end{cases}
\label{eq:phaseSep}
\end{align}
where \(\Omega \subset \mathbb{R}^N\), \(1 \le N \le 3\) is a bounded Lipschitz domain, \(\phi\) a convex function, the kernel \(\mathcal{K}\) represents non-local attracting forces, and \(w\) and \(v\) are the interaction and chemical potentials.
The initial value \(u_0\) satisfies \(0 < u_0 < 1\) and the system is completed with homogeneous Neumann boundary conditions.
A short-time version of this model can be used for image segmentation and contrast enhancement.

Phase separation: starting at the smallest scale Phase separation: the larger reference scale

A non-linear eigenvalue equation
\begin{align*}
- \nabla \cdot \left( \lvert \nabla u \rvert^{p-2} \nabla u \right)
= F_p \lvert u \rvert^{p-2} u ~\text{ in $\Omega$}
\end{align*}
with \( \nu \cdot \nabla u = 0 \) on \( \partial \Omega \)
defines an eigenvalue \(F_p\) with the meaning of the minimal surface halving
the volume.

If applied to the graph metric "every edge of size 1", it is equipartitioning
the graph with the minimal number of the cut edges.

Nearly ten years of work went into the parallel direct linear solver
PARDISO.

Solving linear systems on NEC Vector Engine (work in progress)

The new Vector Engine NEC SX Aurora TSUBASA has 8 cores,
24 or 48 GB of HBM2 on chip and can execute 32 double precision
floating point operations per cycle and has vector registers for
256 floating point values, resulting in a 16,384 bit vector width
and a peak performance of up to 2.45 TFLOPS.

This large vector width requires a redesign of the data structure
in order to efficiently utilize the full power of the Vector Engine.
Moreover, the high bandwidth of the HBM2 allows effective
use of iterative solvers.