Numerical convergence for semilinear parabolic equations

We present a convergence result for finite element discretisations of semilinear parabolic equations, in which the evaluation of the nonlinearity requires some high order of regularity of the solution. For example a coefficient might depend on derivatives or pointevaluation of the solution. We do not rely on high regularity of the exact solution itself and as a payoff we can not deduce convergence rates. As an example the convergence result is applied to a nonlinear Fokker--Planck type battery model

Stationary solutions of liquid two-layer thin film models

We investigate stationary solutions of a thin-film model for liquid two-layer flows in an energetic formulation that is motivated by its gradient flow structure. The goal is to achieve a rigorous understanding of the contact-angle conditions for such two-layer systems. We pursue this by investigating a corresponding energy that favors the upper liquid to dewet from the lower liquid substrate, leaving behind a layer of thickness $h_*$. After proving existence of stationary solutions for the resulting system of thin-film equations we focus on the limit $h_*\to 0$ via matched asymptotic analysis. This yields a corresponding sharp-interface model and a matched asymptotic solution that includes logarithmic switch-back terms. We compare this with results obtained using $\Gamma$-convergence, where we establish existence and uniqueness of energetic minimizers in that limit

Assessment of the potential impacts of plant traits across environments by combining global sensitivity analysis and dynamic modeling in wheat

A crop can be viewed as a complex system with outputs (e.g. yield) that are affected by inputs of genetic, physiology, pedo-climatic and management information. Application of numerical methods for model exploration assist in evaluating the major most influential inputs, providing the simulation model is a credible description of the biological system. A sensitivity analysis was used to assess the simulated impact on yield of a suite of traits involved in major processes of crop growth and development, and to evaluate how the simulated value of such traits varies across environments and in relation to other traits (which can be interpreted as a virtual change in genetic background). The study focused on wheat in Australia, with an emphasis on adaptation to low rainfall conditions. A large set of traits (90) was evaluated in a wide target population of environments (4 sites x 125 years), management practices (3 sowing dates x 2 N fertilization) and $CO_2$ (2 levels). The Morris sensitivity analysis method was used to sample the parameter space and reduce computational requirements, while maintaining a realistic representation of the targeted trait x environment x management landscape ($\sim$ 82 million individual simulations in total). The patterns of parameter x environment x management interactions were investigated for the most influential parameters, considering a potential genetic range of +/- 20% compared to a reference. Main (i.e. linear) and interaction (i.e. non-linear and interaction) sensitivity indices calculated for most of APSIM-Wheat parameters allowed the identifcation of 42 parameters substantially impacting yield in most target environments. Among these, a subset of parameters related to phenology, resource acquisition, resource use efficiency and biomass allocation were identified as potential candidates for crop (and model) improvement.Comment: 22 pages, 8 figures. This work has been submitted to PLoS On

The behavior of a many particle cathode in a lithium-ion battery

We study the almost reversible storage process of charging and discharging of lithium-ion batteries. That process is accompanied by a phase transition and charging and discharging run along different paths, so that hysteretic behavior is observed. We are interested in the storage problem of the cathode of a lithium-ion battery consisting of a system of many iron phosphate (FePO4) particles. There are mathematical models, see [DGJ08], [DGGHJ09] and [DG09], that describe phase transitions and hysteresis exclusively in a single storage particle and they can describe the observed hysteretic voltage-charge plots with almost horizontal plateaus. Interestingly the models predict that the coexistence of a 2-phase system in an individual particle disappears, if its size is below a critical value. The disappearance of the phase transition in the single particle model implies the disappearance of the hysteresis. However, in the experiment hysteretic behavior survives. In other words: The behavior of a storage system consisting of many particles is qualitatively independent of the fact whether the individual particles itself develop a 2-phase system or if they remain in a single phase state. This apparent paradoxical observation will be resolved in this article by a many particle model. It will be shown that if each of the individual particles is in a homogeneous state, nevertheless the many particle ensemble exhibits phase transition and hysteresis ..

Hysteresis in the context of hydrogen storage and lithium-ion batteries

The processes of reversible storage of hydrogen in a metal by loading and unloading and of charging and discharging of lithium-ion batteries have many things in common. The both processes are accompanied by a phase transition and loading and unloading run along different paths, so that hysteretic behavior is observed. For hydrogen storage we consider a fine powder of magnesium (Mg) particles and lithium storage is studied for iron phosphate (FePO$_4$) particles forming the cathode of a lithium-ion battery. The mathematical models that are established in \cite{DGJ08} and \cite{DGH09a}, describe phase transitions and hysteresis exclusively in a single particle and on that basis they can predict the observed hysteretic plots with almost horizontal plateaus. Interestingly the models predict that the coexistence of a 2-phase system in an individual particle disappears, if its size is below a critical value. However, measurements reveal that this is qualitatively not reflected by the mentioned hysteretic plots of loading and unloading. In other words: The behavior of a storage system consisting of many particles is qualitatively independent of the fact whether the individual particles itself develop a 2-phase system or if they remain in a single phase state. This apparent paradoxical observation will be resolved in this article. It will be shown that if each of the individual particles homogeneously distributes the supplied matter, nevertheless the many particle ensemble exhibits phase transition and hysteresis, because one of the two phases is realized in some part of the particles while the remaining part is in the other phase

Gradient flow perspective of thin-film bilayer flows

We study gradient flow formulations of thin-film bilayer flows with triple-junctions between liquid/liquid/air. First we highlight the gradient structure in the Stokes free-boundary flow and identify its solutions with the well known PDE with boundary conditions. Next we propose a similar gradient formulation for the corresponding thin-film model and formally identify solutions with those of the corresponding free-boundary problem. A robust numerical algorithm for the thin-film gradient flow structure is then provided. Using this algorithm we compare the sharp triple-junction model with precursor models. For their stationary solutions a rigorous connection is established using Gamma-convergence. For time-dependent solutions the comparison of numerical solutions shows a good agreement for small and moderate times. Finally we study spreading in the zero-contact angle case, where we compare numerical solutions with asymptotically exact source-type solutions

Gradient flow perspective of thin-film bilayer flows

We study gradient flow formulations of thin-film bilayer flows with triple-junctions between liquid/liquid/air. First we highlight the gradient structure in the Stokes free-boundary flow and identify its solutions with the well-known PDE with boundary conditions. Next we propose a similar gradient formulation for the corresponding thin-film model and formally identify solutions with those of the corresponding free-boundary problem. A robust numerical algorithm for the thin-film gradient flow structure is then provided. Using this algorithm we compare the sharp triple-junction model with precursor models. For their stationary solutions a rigorous connection is established using [Gamma]-convergence. For time-dependentsolutions the comparison of numerical solutions shows a good agreement for small and moderate times. Finally we study spreading in the zero-contact angle case, where we compare numerical solutions with asymptotically exact source-type solutions

Stationary solutions for two-layer lubrication equations

We investigate stationary solutions of flows of thin liquid bilayers in an energetic formulation which is motivated by the gradient flow structure of its lubrication approximation. The corresponding energy favors the liquid substrate to be only partially covered by the upper liquid. This is expressed by a negative spreading coefficient which arises from an intermolecular potential combining attractive and repulsive forces and leads to an ultra-thin layer of thickness ε. For the corresponding lubrication models existence of stationary solutions is proven. In the limit ε to 0 matched asymptotic analysis is applied to derive sharp-interface models and the corresponding contact angles, i.e. the Neumann triangle. In addition we use Γ-convergence and derive the equivalent sharp-interface models rigorously in this limit. For the resulting model existence and uniqueness of energetic minimizers are proven. The minimizers agree with solutions obtained by matched asymptotics

Blow-up versus boundedness in a nonlocal and nonlinear Fokker-Planck equation

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Blow-up versus boundedness in a nonlocal and nonlinear Fokker--Planck equation

We consider a Fokker-Planck equation on a compact interval where, as a constraint, the first moment is a prescribed function of time. Eliminating the associated Lagrange multiplier one obtains nonlinear and nonlocal terms. After establishing suitable local existence results, we use the relative entropy as an energy functional. However, the time-dependent constraint leads to a source term such that a delicate analysis is needed to show that the dissipation terms are strong enough to control the work done by the constraint. We obtain global existence of solutions as long as the prescribed first moment stays in the interior of an interval. If the prescribed moment converges to a constant value inside the interior of the interval, then the solution stabilises to the unique steady state