Guaranteed and robust a posteriori error estimates for singularly perturbed reaction-diffusion problems

International audienceWe derive a posteriori error estimates for singularly perturbed reaction-diffusion problems which yield a guaranteed upper bound on the discretization error and are fully and easily computable. Moreover, they are also locally efficient and robust in the sense that they represent local lower bounds for the actual error, up to a generic constant independent in particular of the reaction coefficient. We present our results in the framework of the vertex-centered finite volume method but their nature is general for any conforming method, like the piecewise linear finite element one. Our estimates are based on a H(div)-conforming reconstruction of the diffusive flux in the lowest-order Raviart-Thomas space linked with mesh dual to the original simplicial one, previously introduced by the last author in the pure diffusion case. They also rely on elaborated Poincaré, Friedrichs, and trace inequalities-based auxiliary estimates designed to cope optimally with the reaction dominance. In order to bring down the ratio of the estimated and actual overall energy error as close as possible to the optimal value of one, independently of the size of the reaction coefficient, we finally develop the ideas of local minimizations of the estimators by local modifications of the reconstructed diffusive flux. The numerical experiments presented confirm the guaranteed upper bound, robustness, and excellent efficiency of the derived estimates

Colloquium: Mechanical formalisms for tissue dynamics

The understanding of morphogenesis in living organisms has been renewed by tremendous progressin experimental techniques that provide access to cell-scale, quantitative information both on theshapes of cells within tissues and on the genes being expressed. This information suggests that ourunderstanding of the respective contributions of gene expression and mechanics, and of their crucialentanglement, will soon leap forward. Biomechanics increasingly benefits from models, which assistthe design and interpretation of experiments, point out the main ingredients and assumptions, andultimately lead to predictions. The newly accessible local information thus calls for a reflectionon how to select suitable classes of mechanical models. We review both mechanical ingredientssuggested by the current knowledge of tissue behaviour, and modelling methods that can helpgenerate a rheological diagram or a constitutive equation. We distinguish cell scale ("intra-cell")and tissue scale ("inter-cell") contributions. We recall the mathematical framework developpedfor continuum materials and explain how to transform a constitutive equation into a set of partialdifferential equations amenable to numerical resolution. We show that when plastic behaviour isrelevant, the dissipation function formalism appears appropriate to generate constitutive equations;its variational nature facilitates numerical implementation, and we discuss adaptations needed in thecase of large deformations. The present article gathers theoretical methods that can readily enhancethe significance of the data to be extracted from recent or future high throughput biomechanicalexperiments.Comment: 33 pages, 20 figures. This version (26 Sept. 2015) contains a few corrections to the published version, all in Appendix D.2 devoted to large deformation

Using the past to constrain the future: how the palaeorecord can improve estimates of global warming

Climate sensitivity is defined as the change in global mean equilibrium temperature after a doubling of atmospheric CO2 concentration and provides a simple measure of global warming. An early estimate of climate sensitivity, 1.5-4.5{\deg}C, has changed little subsequently, including the latest assessment by the Intergovernmental Panel on Climate Change. The persistence of such large uncertainties in this simple measure casts doubt on our understanding of the mechanisms of climate change and our ability to predict the response of the climate system to future perturbations. This has motivated continued attempts to constrain the range with climate data, alone or in conjunction with models. The majority of studies use data from the instrumental period (post-1850) but recent work has made use of information about the large climate changes experienced in the geological past. In this review, we first outline approaches that estimate climate sensitivity using instrumental climate observations and then summarise attempts to use the record of climate change on geological timescales. We examine the limitations of these studies and suggest ways in which the power of the palaeoclimate record could be better used to reduce uncertainties in our predictions of climate sensitivity.Comment: The final, definitive version of this paper has been published in Progress in Physical Geography, 31(5), 2007 by SAGE Publications Ltd, All rights reserved. \c{opyright} 2007 Edwards, Crucifix and Harriso

Flow of foam through a convergent channel

International audienceWe study experimentally the flow of a foam confined as a bubble monolayer between two plates through a convergent channel. We quantify the velocity, the distribution and orientation of plastic events, and the elastic stress, using image analysis. We use two different soap solutions: a sodium dodecyl sulfate (SDS) solution, with a negligible wall friction between the bubbles and the confining plates, and a mixture containing a fatty acid, giving a large wall friction. We show that for SDS solutions, the velocity profile obeys a self-similar form which results from the superposition of plastic events, and the elastic deformation is uniform. For the other solution, the velocity field differs and the elastic deformation increases towards the exit of the channel. We discuss and quantify the role of wall friction on the velocity profile, the elastic deformation, and the rate of plastic events

The ACER pollen and charcoal database: a global resource to document vegetation and fire response to abrupt climate changes during the last glacial period

Quaternary records provide an opportunity to examine the nature of the vegetation and fire responses to rapid past climate changes comparable in velocity and magnitude to those expected in the 21st-century. The best documented examples of rapid climate change in the past are the warming events associated with the Dansgaard–Oeschger (D–O) cycles during the last glacial period, which were sufficiently large to have had a potential feedback through changes in albedo and greenhouse gas emissions on climate. Previous reconstructions of vegetation and fire changes during the D–O cycles used independently constructed age models, making it difficult to compare the changes between different sites and regions. Here, we present the ACER (Abrupt Climate Changes and Environmental Responses) global database, which includes 93 pollen records from the last glacial period (73–15 ka) with a temporal resolution better than 1000 years, 32 of which also provide charcoal records. A harmonized and consistent chronology based on radiometric dating (14C, 234U∕230Th, optically stimulated luminescence (OSL), 40Ar∕39Ar-dated tephra layers) has been constructed for 86 of these records, although in some cases additional information was derived using common control points based on event stratigraphy. The ACER database compiles metadata including geospatial and dating information, pollen and charcoal counts, and pollen percentages of the characteristic biomes and is archived in Microsoft AccessTM at https://doi.org/10.1594/PANGAEA.870867

Mediterranean winter rainfall in phase with African monsoons during the past 1.36 million years

Mediterranean climates are characterized by strong seasonal contrasts between dry summers and wet winters. Changes in winter rainfall are critical for regional socioeconomic development, but are difficult to simulate accurately1 and reconstruct on Quaternary timescales. This is partly because regional hydroclimate records that cover multiple glacial–interglacial cycles2,3 with different orbital geometries, global ice volume and atmospheric greenhouse gas concentrations are scarce. Moreover, the underlying mechanisms of change and their persistence remain unexplored. Here we show that, over the past 1.36 million years, wet winters in the northcentral Mediterranean tend to occur with high contrasts in local, seasonal insolation and a vigorous African summer monsoon. Our proxy time series from Lake Ohrid on the Balkan Peninsula, together with a 784,000-year transient climate model hindcast, suggest that increased sea surface temperatures amplify local cyclone development and refuel North Atlantic low-pressure systems that enter the Mediterranean during phases of low continental ice volume and high concentrations of atmospheric greenhouse gases. A comparison with modern reanalysis data shows that current drivers of the amount of rainfall in the Mediterranean share some similarities to those that drive the reconstructed increases in precipitation. Our data cover multiple insolation maxima and are therefore an important benchmark for testing climate model performance