Robust a priori and a posteriori error analysis for the approximation of Allen–Cahn and Ginzburg–Landau equations past topological changes

A priori and a posteriori error estimates are derived for the numerical approximation of scalar and complex valued phase field models. Particular attention is devoted to the dependence of the estimates on a small parameter and to the validity of the estimates in the presence of topological changes in the solution that represents singular points in the evolution. For typical singularities the estimates depend on the inverse of the parameter in a polynomial as opposed to exponential dependence of estimates resulting from a straightforward error analysis. The estimates naturally lead to adaptive mesh refinement and coarsening algorithms. Numerical experiments illustrate the reliability and efficiency of this approach for the evolution of interfaces and vortices that undergo topological changes

Modern spatial sea-ice variability in the central Arctic Ocean and adjacent marginal seas: Reconstruction from biomarker data

Sea ice is a fundamental component of Earth’s climate system, contributing to heat reduction (albedo) and deep-water formation. In order to understand processes controlling the recent dramatic reduction in Arctic sea-ice cover, it is essential to determine spatial and temporal changes in sea-ice occurrence and its natural variability in the present and past. Here, we present biomarker data from surface sediments and related to the modern spatial (seasonal) sea-ice variability in the central Arctic Ocean and adjacent marginal seas (i.e., Bering, Chukchi, Laptev and Kara seas) as well as the Fram Strait/Yermak Plateau area. We determined concentrations of the sea-ice diatom-derived biomarker “IP25″ (highly-branched isoprenoid – HBI – with 25 carbon atom; Belt et al., 2007), phytoplankton-derived biomarkers (brassicasterol and dinosterol) and terrigenous biomarkers (campesterol and Î_-sitosterol) to estimate recent sea-ice conditions in the study area. A combined phytoplankton-IP25 biomarker approach (“PIP25 index”; Müller et al., 2009, 2011) is used to reconstruct the modern sea-ice distribution more quantitatively. In addition, the distribution pattern of HBI-diene/IP25 ratios has been determined to test a proposed relationship between the diene/IP25 ratio and sea-surface temperatures in Arctic marginal ice-zone environments (Fahl and Stein, 2012; Stein et al., 2012). Assessment of sea-ice conditions based on these biomarker data display that a quite stable marginal ice zone exists along the continental shelf/slope of Kara and Laptev seas during summer/early fall. Elevated IP25 as well as brassicasterol and dinosterol values occurring in the central Kara and Laptev seas are related to extended sea-ice-cover and higher primary production (close to ice-edge situation). Further to the north and the central Arctic Ocean, lower IP25 and phytoplankton biomarker concentrations point to a more close sea-ice cover situation

Wind and sky as compass cues in desert ant navigation

While integrating their foraging and homing paths, desert ants, Cataglyphis fortis, depend on external compass cues. Whereas recent research in bees and ants has focused nearly exclusively on the polarization compass, two other compass systems—the sun compass and the wind (anemo) compass—as well as the mutual interactions of all these compass systems have received little attention. In this study, we show that of the two visual compass systems, it is only the polarization compass that invariably outcompetes the wind compass, while the sun compass does so only under certain conditions. If the ants are experimentally deprived of their polarization compass system, but have access simultaneously to both their sun compass and their wind compass, they steer intermediate courses. The intermediate courses shift the more towards the wind compass course, the higher the elevation of the sun is in the sk

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

A posteriori error control for stationary coupled bulk-surface equations

We consider a system of two coupled elliptic equations, one defined on a bulk domain and the other one on the boundary surface. Problems of this kind are relevant for applications in engineering, chemistry and in biology like e.g. biological signal transduction. For the a posteriori error control of the coupled system, a residual error estimator is derived which takes into account the approximation errors due to the finite element discretisation in space as well as the polyhedral approximation of the surface. An adaptive refinement algorithm controls the overall error. Numerical experiments illustrate the performance of the a posteriori error estimator and the adaptive algorithm with several benchmark examples

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 posteriori error control for stationary coupled bulk-surface equations

We consider a system of two coupled elliptic equations, one defined on a bulk domain and the other one on the boundary surface. Problems of this kind of problem are relevant for applications in engineering, chemistry and in biology like e.g. biological signal transduction. For the a posteriori error control of the coupled system, a residual error estimator is derived which takes into account the approximation errors due to the finite element discretisation in space as well as the polyhedral approximation of the surface. An adaptive refinement algorithm controls the overall error. Numerical experiments illustrate the performance of the a posteriori error estimator and the proposed adaptive algorithm with several benchmark examples

Thermodynamic models for a concentration and electric field dependent susceptibility in liquid electrolytes

The dielectric susceptibility $chi$ is an elementary quantity of the electrochemical double layer and the associated Poisson equation. While most often $chi$ is treated as a material constant, its dependency on the salt concentration in liquid electrolytes is demonstrated by various bulk electrolyte experiments. This is usually referred to as dielectric decrement. Further, it is theoretically well accepted that the susceptibility declines for large electric fields. This effect is frequently termed dielectric saturation. We analyze the impact of a variable susceptibility in terms of species concentrations and electric fields based on non-equilibrium thermodynamics. This reveals some non-obvious generalizations compared to the case of a constant susceptibility. In particular the consistent coupling of the Poisson equation, the momentum balance and the chemical potentials functions are of ultimate importance. In a numerical study, we systematically analyze the effects of a concentration and field dependent susceptibility on the double layer of a planar electrode electrolyte interface. We compute the differential capacitance and the spatial structure of the electric potential, solvent concentration and ionic distribution for various non-constant models of $chi$