65M08 Finite volume methods
Large-area organic light-emitting diodes are thin-film multilayer devices that show pronounced self-heating and brightness inhomogeneities at high currents. As these high currents are typical for lighting applications, a deeper understanding of the mechanisms causing these inhomogeneities is necessary. We discuss the modeling of the interplay between current flow, self-heating, and heat transfer in such devices using a system of partial differential equations of thermistor type, that is capable of explaining the development of luminance inhomogeneities. The system is based on the heat equation for the temperature coupled to a p(x)-Laplace-type equation for the electrostatic potential with mixed boundary conditions. The p(x)-Laplacian allows to take into account non-Ohmic electrical behavior of the different organic layers. Moreover, we present analytical results on the existence, boundedness, and regularity of solutions to the system. A numerical scheme based on the finite-volume method allows for efficient simulations of device structures.
In large-area Organic Light-Emitting Diodes (OLEDs) spatially inhomogeneous luminance at high power due to inhomogeneous current flow and electrothermal feedback can be observed. To describe these self-heating effects in organic semiconductors we present a stationary thermistor model based on the heat equation for the temperature coupled to a p-Laplace-type equation for the electrostatic potential with mixed boundary conditions. The p-Laplacian describes the non-Ohmic electrical behavior of the organic material. Moreover, an Arrhenius-like temperature dependency of the electrical conductivity is considered.
We introduce a finite-volume scheme for the system and discuss its relation to recent network models for OLEDs. In two spatial dimensions we derive a priori estimates for the temperature and the electrostatic potential and prove the existence of a weak solution by Schauder's fixed point theorem.
We study the approximation of Wasserstein gradient structures
by their finite-dimensional analog. We show that simple
finite-volume discretizations of the linear Fokker-Planck
equation exhibit the recently established entropic gradient-flow
structure for reversible Markov chains. Then we reprove the
convergence of the discrete scheme in the limit of vanishing
mesh size using only the involved gradient-flow structures.
In particular, we make no use of the linearity of the equations
nor of the fact that the Fokker-Planck equation is of second order.
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.
We consider discretizations for reaction-diffusion systems with nonlinear
diffusion in two space dimensions. The applied model allows to handle heterogeneous
materials and uses the chemical potentials of the involved species as primary variables.
We propose an implicit Voronoi finite volume discretization on regular Delaunay
meshes that allows to prove uniform, mesh-independent global upper and lower L1
bounds for the chemical potentials. These bounds provide the main step for a convergence
analysis for the full discretized nonlinear evolution problem. The fundamental
ideas are energy estimates, a discrete Moser iteration and the use of discrete
Gagliardo-Nirenberg inequalities. For the proof of the Gagliardo-Nirenberg inequalities
we exploit that the discrete Voronoi finite volume gradient norm in 2d coincides
with the gradient norm of continuous piecewise linear finite elements.