35R05 Partial differential equations with discontinuous coefficients or data
We investigate the convergence of an implicit Voronoi finite volume
method for reaction-diffusion problems including nonlinear diffusion
in two space dimensions. The model allows to handle heterogeneous
materials and uses the chemical potentials of the involved species as
primary variables. The numerical scheme uses boundary conforming Delaunay
meshes and preserves positivity and the dissipative property of the
continuous system. Starting from a result on the global stability of
the scheme (uniform, mesh-independent global upper and lower bounds),
we prove strong convergence of the chemical activities and their gradients
to a weak solution of the continuous problem. In order to illustrate the
preservation of qualitative properties by the numerical scheme, we present
a long-term simulation of the Michaelis-Menten-Henri system. Especially,
we investigate the decay properties of the relative free energy and the
evolution of the dissipation rate over several magnitudes of time, and
obtain experimental orders of convergence for these quantities.
An electronic model for solar cells including active interfaces and energy resolved defect densities
(2011)
We introduce an electronic model for solar cells taking into account
heterostructures with active
interfaces and energy resolved volume and interface trap densities.
The model consists of continuity equations for electrons and holes with thermionic
emission transfer conditions at the interface and of ODEs for the trap
densities with energy level and spatial position as parameters,
where the right hand sides contain generation-recombination as well as
ionization reactions. This system is coupled with a Poisson
equation for the electrostatic potential.
We show the thermodynamic correctness of the model and prove a priori estimates
for the solutions to the evolution system. Moreover, existence and uniqueness
of weak solutions of the problem are proven. For this purpose we solve a
regularized problem and verify bounds of the corresponding solution
not depending on the regularization level.
We investigate optimal elliptic
regularity (within the scale of Sobolev spaces) of anisotropic
div--grad operators in three dimensions at a multi-material vertex on
the Neumann boundary part of a polyhedral spatial domain. The
gradient of a solution to the corresponding elliptic PDE (in a
neighbourhood of the vertex) is integrable to an index greater than
three.
We introduce an electronic model for solar cells including energy resolved defect
densities. The resulting drift-diffusion model corresponds to a generalized
van Roosbroeck system with additional source terms coupled with ODEs containing space and
energy as parameters for all defect densities. The system has to be considered in
heterostructures and with mixed boundary conditions from device simulation.
We give a weak formulation of the problem. If the boundary data and the sources
are compatible with thermodynamic equilibrium the free energy along solutions
decays monotonously. In other cases it may be increasing, but we estimate its growth.
We establish boundedness and uniqueness results and prove the existence of a
weak solution. This is done by considering a regularized problem, showing its
solvability and the boundedness of its solutions independent of the regularization level.
Local existence, uniqueness and smooth dependence for nonsmooth quasilinear parabolic problems
(2009)
We prove local existence, uniqueness, Hölder regularity in space and time, and smooth dependence in Hölder spaces for a general class of quasilinear parabolic initial boundary value problems with nonsmooth data.
As a result the gap between low smoothness of the data, which is typical for
many applications, and high smoothness of the solutions, which is necessary
for the applicability of differential calculus to abstract formulations of the
initial boundary value problems, has been closed. The theory works for any
space dimension, and the nonlinearities are allowed to be nonlocal and to
have any growth. The main tools are new maximal regularity results [19, 20]
in Sobolev–Morrey spaces for linear parabolic initial boundary value problems
with nonsmooth data, linearization techniques and the Implicit Function Theorem.
Let $\Upsilon$ be a three-dimensional Lipschitz polyhedron, and assume that the matrix function $\mu$ is piecewise constant on a polyhedral partition of $\Upsilon$. Based on regularity results for solutions to two-dimensional anisotropic transmission problems near corner points we obtain conditions on $\mu$ and the intersection angles between interfaces and $\partial \Upsilon$ ensuring that the operator $-\nabla \cdot \mu \nabla$ maps the Sobolev space $W^1,q_0(\Upsilon)$ isomorphically onto $W^-1,q(\Upsilon)$ for some $q > 3$.