A finite element method for fully nonlinear elliptic problems

We present a continuous finite element method for some examples of fully nonlinear elliptic equation. A key tool is the discretisation proposed in Lakkis & Pryer (2011, SISC) allowing us to work directly on the strong form of a linear PDE. An added benefit to making use of this discretisation method is that a recovered (finite element) Hessian is a biproduct of the solution process. We build on the linear basis and ultimately construct two different methodologies for the solution of second order fully nonlinear PDEs. Benchmark numerical results illustrate the convergence properties of the scheme for some test problems including the Monge-Amp\`ere equation and Pucci's equation.Comment: 22 pages, 31 figure

Exhibiting cross-diffusion-induced patterns for reaction-diffusion systems on evolving domains and surfaces

The aim of this manuscript is to present for the first time the application of the finite element method for solving reaction-diffusion systems with cross-diffusion on continuously evolving domains and surfaces. Furthermore we present pattern formation generated by the reaction-diffusion systemwith cross-diffusion on evolving domains and surfaces. A two-component reaction-diffusion system with linear cross-diffusion in both u and v is presented. The finite element method is based on the approximation of the domain or surface by a triangulated domain or surface consisting of a union of triangles. For surfaces, the vertices of the triangulation lie on the continuous surface. A finite element space of functions is then defined by taking the continuous functions which are linear affine on each simplex of the triangulated domain or surface. To demonstrate the role of cross-diffusion to the theory of pattern formation, we compute patterns with model kinetic parameter values that belong only to the cross-diffusion parameter space; these do not belong to the standard parameter space for classical reaction-diffusion systems. Numerical results exhibited show the robustness, flexibility, versatility, and generality of our methodology; the methodology can deal with complicated evolution laws of the domain and surface, and these include uniform isotropic and anisotropic growth profiles as well as those profiles driven by chemical concentrations residing in the domain or on the surface

Spatial variation in boundary conditions can govern selection and location of eyespots in butterfly wings

Despite being the subject of widespread study, many aspects of the development of eyespot patterns in butterfly wings remain poorly understood. In this work, we examine, through numerical simulations, a mathematical model for eyespot focus point formation in which a reaction-diffusion system is assumed to play the role of the patterning mechanism. In the model, changes in the boundary conditions at the veins at the proximal boundary alone are capable of determining whether or not an eyespot focus forms in a given wing cell and the eventual position of focus points within the wing cell. Furthermore, an auxiliary surface reaction diffusion system posed along the entire proximal boundary of the wing cells is proposed as the mechanism that generates the necessary changes in the proximal boundary profiles. In order to illustrate the robustness of the model, we perform simulations on a curved wing geometry that is somewhat closer to a biological realistic domain than the rectangular wing cells previously considered, and we also illustrate the ability of the model to reproduce experimental results on artificial selection of eyespots.Publisher PD

An adaptive finite element method for the infinity Laplacian

We construct a finite element method (FEM) for the infinity Laplacian. Solutions of this problem are well known to be singular in nature so we have taken the opportunity to conduct an a posteriori analysis of the method deriving residual based estimators to drive an adaptive algorithm. It is numerically shown that optimal convergence rates are regained using the adaptive procedure

Diferencias estacionales entre tropopausas térmicas calculadas usando perfiles verticales de temperatura a partir de niveles significativos y estándar de radiosondeos para la red aerológica de Sudamérica

Se llevó a cabo un estudio de diferencias entre dos tropopausas térmicas sobre 59 estaciones aerológicas de Sudamérica. La tropopausa térmica primigenia, llamada lapse rate tropopause (LRT) y obtenida de perfiles verticales de temperatura en niveles significativos incluidos en los radiosondeos siguiendo la definición de la Organización Meteorológica Mundial (World Meteorological Organization, WMO), fue comparada con la tropopausa estimada de los niveles estándar (LRTM) con un algoritmo que está de acuerdo con la definición de la WMO. El período de estudio comprende enero 1973‒diciembre 2019. Las comparaciones entre la LRT y la LRTM se llevaron a cabo de manera estacional para presión, altura y temperatura. Los resultados fueron organizados en bandas latitudinales de 5º de ancho. Por bandas zonales la LRTM está, en promedio, por debajo o por encima de la LRT dependiendo de la banda, y en general ambas tropopausas son estadísticamente distintas. Con el objetivo de estimar si existe una relación entre ellas se calculó el coeficiente de correlación de Spearman entre las mismas variables de LRTM y LRT, los cuales resultaron positivos por lo que las variables correlacionadas pueden vincularse a través de funciones monótonas crecientes. Se escogió una función lineal y las pendientes y ordenadas al origen fueron obtenidas para cada banda de latitud. Las variables de la LRTM tomadas como predictores permiten así conocer las correspondientes a la LRT estimada (LRT*). La discrepancia entre LRT* y LRT fueron cuantificadas con el error cuadrático medio, cuya maximización tiene lugar en la región de influencia del jet subtropical.An analysis on the differences between two thermal tropopauses at 59 South American upper-air stations was carried out. The original lapse rate tropopause (LRT) derived from significant-level radiosonde temperature profiles following the World Meteorological Organization (WMO) definition was compared with the tropopause estimated from radiosonde mandatory levels (LRTM) with a WMO-compliant algorithm. The study period encompasses January 1973–December 2019. The comparisons between the LRT and the LRTM were performed on a seasonal basis for pressure, height and temperature and the results were organized into latitudinal bands of 5° width. On average, the LRTM locates below or above the LRT, depending on the band considered, and in general the both tropopauses are statistically different. The Spearman’s correlation coefficients between the same LRTM and LRT variables were calculated in order to estimate whether there exists a relation between them. The coefficients resulted positive so the correlated variables can be related through monotonic increasing functions. A linear function was selected and the slopes and the intercepts for this relation were provided at each latitudinal band. Taking the LRTM variables as predictors enables the obtention of the corresponding LRT variables. The matching between the LRT and the LRTM-derived LRT was addressed through root mean squares, whose maximization occurs in the subtropical jet’s region of influence.Asociación Argentina de Geofísicos y Geodesta

Diferencias estacionales entre tropopausas térmicas calculadas usando perfiles verticales de temperatura a partir de niveles significativos y estándar de radiosondeos para la red aerológica de Sudamérica

Se llevó a cabo un estudio de diferencias entre dos tropopausas térmicas sobre 59 estaciones aerológicas de Sudamérica. La tropopausa térmica primigenia, llamada lapse rate tropopause (LRT) y obtenida de perfiles verticales de temperatura en niveles significativos incluidos en los radiosondeos siguiendo la definición de la Organización Meteorológica Mundial (World Meteorological Organization, WMO), fue comparada con la tropopausa estimada de los niveles estándar (LRTM) con un algoritmo que está de acuerdo con la definición de la WMO. El período de estudio comprende enero 1973‒diciembre 2019. Las comparaciones entre la LRT y la LRTM se llevaron a cabo de manera estacional para presión, altura y temperatura. Los resultados fueron organizados en bandas latitudinales de 5º de ancho. Por bandas zonales la LRTM está, en promedio, por debajo o por encima de la LRT dependiendo de la banda, y en general ambas tropopausas son estadísticamente distintas. Con el objetivo de estimar si existe una relación entre ellas se calculó el coeficiente de correlación de Spearman entre las mismas variables de LRTM y LRT, los cuales resultaron positivos por lo que las variables correlacionadas pueden vincularse a través de funciones monótonas crecientes. Se escogió una función lineal y las pendientes y ordenadas al origen fueron obtenidas para cada banda de latitud. Las variables de la LRTM tomadas como predictores permiten así conocer las correspondientes a la LRT estimada (LRT*). La discrepancia entre LRT* y LRT fueron cuantificadas con el error cuadrático medio, cuya maximización tiene lugar en la región de influencia del jet subtropical.An analysis on the differences between two thermal tropopauses at 59 South American upper-air stations was carried out. The original lapse rate tropopause (LRT) derived from significant-level radiosonde temperature profiles following the World Meteorological Organization (WMO) definition was compared with the tropopause estimated from radiosonde mandatory levels (LRTM) with a WMO-compliant algorithm. The study period encompasses January 1973–December 2019. The comparisons between the LRT and the LRTM were performed on a seasonal basis for pressure, height and temperature and the results were organized into latitudinal bands of 5° width. On average, the LRTM locates below or above the LRT, depending on the band considered, and in general the both tropopauses are statistically different. The Spearman’s correlation coefficients between the same LRTM and LRT variables were calculated in order to estimate whether there exists a relation between them. The coefficients resulted positive so the correlated variables can be related through monotonic increasing functions. A linear function was selected and the slopes and the intercepts for this relation were provided at each latitudinal band. Taking the LRTM variables as predictors enables the obtention of the corresponding LRT variables. The matching between the LRT and the LRTM-derived LRT was addressed through root mean squares, whose maximization occurs in the subtropical jet’s region of influence.Asociación Argentina de Geofísicos y Geodesta

A model for selection of eyespots on butterfly wings

The development of eyespots on the wing surface of butterflies of the family Nympalidae is one of the most studied examples of biological pattern formation.However, little is known about the mechanism that determines the number and precise locations of eyespots on the wing. Eyespots develop around signaling centers, called foci, that are located equidistant from wing veins along the midline of a wing cell (an area bounded by veins). A fundamental question that remains unsolved is, why a certain wing cell develops an eyespot, while other wing cells do not. We illustrate that the key to understanding focus point selection may be in the venation system of the wing disc. Our main hypothesis is that changes in morphogen concentration along the proximal boundary veins of wing cells govern focus point selection. Based on previous studies, we focus on a spatially two-dimensional reaction-diffusion system model posed in the interior of each wing cell that describes the formation of focus points. Using finite element based numerical simulations, we demonstrate that variation in the proximal boundary condition is sufficient to robustly select whether an eyespot focus point forms in otherwise identical wing cells. We also illustrate that this behavior is robust to small perturbations in the parameters and geometry and moderate levels of noise. Hence, we suggest that an anterior-posterior pattern of morphogen concentration along the proximal vein may be the main determinant of the distribution of focus points on the wing surface. In order to complete our model, we propose a two stage reaction-diffusion system model, in which an one-dimensional surface reaction-diffusion system, posed on the proximal vein, generates the morphogen concentrations that act as non-homogeneous Dirichlet (i.e., fixed) boundary conditions for the two-dimensional reaction-diffusion model posed in the wing cells. The two-stage model appears capable of generating focus point distributions observed in nature. We therefore conclude that changes in the proximal boundary conditions are sufficient to explain the empirically observed distribution of eyespot focus points on the entire wing surface. The model predicts, subject to experimental verification, that the source strength of the activator at the proximal boundary should be lower in wing cells in which focus points form than in those that lack focus points. The model suggests that the number and locations of eyespot foci on the wing disc could be largely controlled by two kinds of gradients along two different directions, that is, the first one is the gradient in spatially varying parameters such as the reaction rate along the anterior-posterior direction on the proximal boundary of the wing cells, and the second one is the gradient in source values of the activator along the veins in the proximal-distal direction of the wing cell

Early 21st century cyclone climatology: a 3D perspective. Basic characterization

Extratropical cyclones are a relevant feature in the climate at middle and high latitudes. Despite their relevance, most of studies typically focus on cyclones identified at a single atmospheric level and on events close to the surface. This paper provides a new perspective on the Southern Hemisphere cyclone events based on the multilevel cyclone tracking algorithm STACKER. The algorithm, using relative vorticity, detects the raw tracks at single levels and objectively combine them to provide the 3D events and their evolutionary timeline. As result, 3D cyclone climatology, based on ECMWF Reanalysis ERA-I data from 12 pressure levels in the troposphere and lowermost stratosphere is presented. To the best of our knowledge this is the first analysis carried out throughout the troposphere and the lowermost extratropical/subpolar stratosphere in order to give a comprehensive picture of the cyclone events as physical entities throughout their lifetime. Cyclone properties analysed are track densities, translational velocity, vorticity and lifetimes. For the subtropical and extrattropical SH, results support many previous ideas about cyclone characteristics, but new insights are also obtained. A total of 58231 multilevel cyclone events lasting at least 2 days were detected, with vertical structures spanning two or more levels. This means an average of 303 cyclone events of all types per month, between 2001 and 2017, disregarding seasonality. Results shows that the lowermost level of cyclones are most frequently detected at 925 and 700 hPa, Considering that cyclonic systems can be grouped into families, results per month on average, show that shallow systems are the most frequent events with approximately 248 systems detected, followed by 43 intermediates and 11 deep events. Shallow and deep systems have a large percentage of events with genesis at 925 and 700hPa. Density statistics show that shallow events are present at all latitude ranges mostly poleward 30ºS with high and medium intensities, while intermediate ones are mostly restricted to mid-latitudes and deep events are mostly confined to sub-polar and polar latitudes. Cyclones over Antarctica seems to be mostly intermediates and deeps, with longer lifetimes and lower velocities

Parameter identification problems in the modelling of cell motility

We present a novel parameter identification algorithm for the estimation of parameters in models of cell motility using imaging data of migrating cells. Two alternative formulations of the objective functional that measures the difference between the computed and observed data are proposed and the parameter identification problem is formulated as a minimisation problem of nonlinear least squares type. A Levenberg–Marquardt based optimisation method is applied to the solution of the minimisation problem and the details of the implementation are discussed. A number of numerical experiments are presented which illustrate the robustness of the algorithm to parameter identification in the presence of large deformations and noisy data and parameter identification in three dimensional models of cell motility. An application to experimental data is also presented in which we seek to identify parameters in a model for the monopolar growth of fission yeast cells using experimental imaging data. Our numerical tests allow us to compare the method with the two different formulations of the objective functional and we conclude that the results with both objective functionals seem to agree