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Optimal and robust a posteriori error estimates in L∞(L2) for the approximation of Allen-Cahn equations past singularities
Optimal a posteriori error estimates in L∞(L2) are derived for the finite element approximation of Allen-Cahn equations. The estimates depend on the inverse of a small parameter only in a low order polynomial and are valid past topological changes of the evolving interface. The error analysis employs an elliptic reconstruction of the approximate solution and applies to a large class of conforming, nonconforming, mixed, and discontinuous Galerkin methods. Numerical experiments illustrate the theoretical results.
Impact of slippage on the morphology and stability of a dewetting rim
In this study lubrication theory is used to describe the stability and morphology of the rim that forms as a thin polymer film dewets from a hydrophobized silicon wafer. Thin film equations are derived from the governing hydrodynamic equations for the polymer to enable the systematic mathematical and numerical analysis of the properties of the solutions for different regimes of slippage and for a range of time scales. Dewetting rates and the cross sectional profiles of the evolving rims are derived for these models and compared to experimental results. Experiments also show that the rim is typically unstable in the spanwise direction and develops thicker and thinner parts that may grow into ``fingers''. Linear stability analysis as well as nonlinear numerical solutions are presented to investigate shape and growth rate of the rim instability. It is demonstrated that the difference in morphology and the rate at which the instability develops can be directly attributed to the magnitude of slippage. Finally, a derivation is given for the dominant wavelength of the bulges along the unstable rim.
A Redlich-Kister type free energy model for Li-intercalation compounds with variable lattice occupation numbers
One of the central quantities of a lithium ion intercalation compound is the open circuit potential, the voltage a battery material delivers in thermodynamic equilibrium. This voltage is related to the chemical potential of lithium in the insertion material and in general a non-linear function of the mole fraction of intercalated lithium. Experimental data shows further that it is specific for various materials. The open circuit voltage is a central ingredient for mathematical models of whole battery cells, which are used to investigate and simulate the charge and discharge behavior and to interpret experimental data on non-equilibrium processes. However, since no overall predictive theoretical method presently exists for the open circuit voltage, it is commonly fitted to experimental data. Simple polynomial fitting approaches are widely used, but they lack any thermodynamic interpretation. More recently systematically and thermodynamically motivated approaches are used to model the open circuit potential. We provide here an explicit free energy density which accounts for variable occupation numbers of Li on the intercalation lattice as well as RedlichKister-type enthalpy contributions. The derived chemical potential is validated by experimental data of Liy(Ni1/3Mn1/3Co1/3)O2 and we show that only two parameters are sufficient to obtain an overall agreement of the non-linear open circuit potential within the experimental error.
Towards a Mathematical Theory of Turbulence in Fluids
Fluid mechanics is the theory of how liquids and gases move around. For the most part, the basic physics are well understood and the mathematical models look relatively simple. Despite this, fluids display a dazzling mystery to their motion. The random-looking, chaotic behavior of fluids is known as turbulence, and it lies far beyond our mathematical understanding, despite a century of intense research.
Spherical arc-length as a global conformal parameter for analytic curves in the Riemann sphere
We prove that for every analytic curve in the complex plane C, Euclidean and spherical arc-lengths are global conformal parameters. We also prove that for any analytic curve in the hyperbolic plane, hyperbolic arc-length is also a global parameter. We generalize some of these results to the case of analytic curves in Rn and Cn and we discuss the situation of curves in the Riemann sphere C {∞}.
Anisotropic finite element mesh adaptation via higher dimensional embedding
In this paper we provide a novel anisotropic mesh adaptation technique for adaptive finite element analysis. It is based on the concept of higher dimensional embedding, which was exploited in [1-4] to obtain an anisotropic curvature adapted mesh that fits a complex surface in ℝ3. In the context of adaptive finite element simulation, the solution (which is an unknown function ƒ: Ω ⊂; ℝd → ℝ) is sought by iteratively modifying a finite element mesh according to a mesh sizing field described via a (discrete) metric tensor field that is typically obtained through an error estimator. We proposed to use a higher dimensional embedding, Φf(x) := (x1, …, xd, s f (x1, …, xd), s ∇ f (x1, …, xd))t, instead of the mesh sizing field for the mesh adaption. This embedding contains both informations of the function ƒ itself and its gradient. An isotropic mesh in this embedded space will correspond to an anisotropic mesh in the actual space, where the mesh elements are stretched and aligned according to the features of the function ƒ. To better capture the anisotropy and gradation of the mesh, it is necessary to balance the contribution of the components in this embedding. We have properly adjusted Φf(x) for adaptive finite element analysis. To better understand and validate the proposed mesh adaptation strategy, we first provide a series of experimental tests for piecewise linear interpolation of known functions. We then applied this approach in an adaptive finite element solution of partial differential equations. Both tests are performed on two-dimensional domains in which adaptive triangular meshes are generated. We compared these results with the ones obtained by the software BAMG - a metric-based adaptive mesh generator. The errors measured in the L2 norm are comparable. Moreover, our meshes captured the anisotropy more accurately than the meshes of BAMG.
Error control for the approximation of Allen-Cahn and Cahn-Hilliard equations with a logarithmic potential
A fully computable upper bound for the finite element approximation error of Allen-Cahn and Cahn-Hilliard equations with logarithmic potentials is derived. Numerical experiments show that for the sharp interface limit this bound is robust past topological changes. Modifications of the abstract results to derive quasi-optimal error estimates in different norms for lowest order finite element methods are discussed and lead to weaker conditions on the residuals under which the conditional error estimates hold.
A Bayesian approach to parameter identification in gas networks
The inverse problem of identifying the friction coefficient in an isothermal semilinear Euler system is considered. Adopting a Bayesian approach, the goal is to identify the distribution of the quantity of interest based on a finite number of noisy measurements of the pressure at the boundaries of the domain. First well-posedness of the underlying non-linear PDE system is shown using semigroup theory, and then Lipschitz continuity of the solution operator with respect to the friction coefficient is established. Based on the Lipschitz property, well-posedness of the resulting Bayesian inverse problem for the identification of the friction coefficient is inferred. Numerical tests for scalar and distributed parameters are performed to validate the theoretical results.
Quantum diffusion
If you place a drop of ink into a glass of water, the ink will slowly dissipate into the surrounding water until it is perfectly mixed. If you record your experiment with a camera and play the film backwards, you will see something that is never observed in the real world. Such diffusive and irreversible behaviour is ubiquitous in nature. Nevertheless, the fundamental equations that describe the motion of individual particles – Newton’s and Schrödinger’s equations – are reversible in time: a film depicting the motion of just a few particles looks as realistic when played forwards as when played backwards. In this snapshot, we discuss how one may try to understand the origin of diffusion starting from the fundamental laws of quantum mechanics.