A posteriori error control for fully discrete Crank–Nicolson schemes

We derive residual-based a posteriori error estimates of optimal order for fully discrete approximations for linear parabolic problems. The time discretization uses the Crank--Nicolson method, and the space discretization uses finite element spaces that are allowed to change in time. The main tool in our analysis is the comparison with an appropriate reconstruction of the discrete solution, which is introduced in the present paper

A posteriori error estimates for fully discrete schemes for the time dependent Stokes problem

The final publication is available at Springer via http://dx.doi.org/10.1007/s10092-018-0259-2This work is devoted to a posteriori error analysis of fully discrete finite element approximations to the time dependent Stokes system. The space discretization is based on popular stable spaces, including Crouzeix–Raviart and Taylor–Hood finite element methods. Implicit Euler is applied for the time discretization. The finite element spaces are allowed to change with time steps and the projection steps include alternatives that is hoped to cope with possible numerical artifices and the loss of the discrete incompressibility of the schemes. The final estimates are of optimal order in L∞(L2) for the velocity error

Convergence of simple adaptive Galerkin schemes based on h − h/2 error estimators

We discuss several adaptive mesh-refinement strategies based on (h − h/2)-error estimation. This class of adaptivemethods is particularly popular in practise since it is problem independent and requires virtually no implementational overhead. We prove that, under the saturation assumption, these adaptive algorithms are convergent. Our framework applies not only to finite element methods, but also yields a first convergence proof for adaptive boundary element schemes. For a finite element model problem, we extend the proposed adaptive scheme and prove convergence even if the saturation assumption fails to hold in general

A variational formulation of anisotropic geometric evolution equations in higher dimensions

Accepted versio

Microdevices for extensional rheometry of low viscosity elastic liquids : a review

Extensional flows and the underlying stability/instability mechanisms are of extreme relevance to the efficient operation of inkjet printing, coating processes and drug delivery systems, as well as for the generation of micro droplets. The development of an extensional rheometer to characterize the extensional properties of low viscosity fluids has therefore stimulated great interest of researchers, particularly in the last decade. Microfluidics has proven to be an extraordinary working platform and different configurations of potential extensional microrheometers have been proposed. In this review, we present an overview of several successful designs, together with a critical assessment of their capabilities and limitations

Edge Detection by Adaptive Splitting II. The Three-Dimensional Case

In Llanas and Lantarón, J. Sci. Comput. 46, 485–518 (2011) we proposed an algorithm (EDAS-d) to approximate the jump discontinuity set of functions defined on subsets of ℝ d . This procedure is based on adaptive splitting of the domain of the function guided by the value of an average integral. The above study was limited to the 1D and 2D versions of the algorithm. In this paper we address the three-dimensional problem. We prove an integral inequality (in the case d=3) which constitutes the basis of EDAS-3. We have performed detailed computational experiments demonstrating effective edge detection in 3D function models with different interface topologies. EDAS-1 and EDAS-2 appealing properties are extensible to the 3D cas

Early‐season mass‐flowering crop cover dilutes wild bee abundance and species richness in temperate regions: A quantitative synthesis

Pollinators benefit from increasing floral resources in agricultural landscapes, which could be an underexplored co‐benefit of mass‐flowering crop cultivation. However, the impacts of mass‐flowering crops on pollinator communities are complex and appear to be context‐dependent, mediated by factors such as crop flowering time and the availability of other flower resources in the landscape. A synthesis of research is needed to develop management recommendations for effective pollinator conservation in agroecosystems. By combining 22 datasets from 13 publications conducted in nine temperate countries (20 European, 2 North American), we investigated if mass‐flowering crop flowering time (early or late season), bloom state (during or after crop flowering) and extent of non‐crop habitat cover in the landscape moderated the effect of mass‐flowering crop cover on wild pollinator abundance and species richness in mass‐flowering crop and non‐crop habitats. During bloom, wild bee abundance and richness are negatively related to mass‐flowering crop cover. Dilution effects were predominant in crop habitats and early in the season, except for bumblebees, which declined with mass‐flowering crop cover irrespective of habitat or season. Late in the season and in non‐crop habitats, several of these negative relationships were either absent or reversed. Late‐season mass‐flowering crop cover is positively related to honeybee abundance in crop habitats and to other bee abundance in non‐crop habitats. These results indicate that crop‐adapted species, like honeybees, move to forage and concentrate on late‐season mass‐flowering crops at a time when flower availability in the landscape is limited, potentially alleviating competition for flower resources in non‐crop habitats. We found no evidence of pollinators moving from mass‐flowering crop to non‐crop habitats after crop bloom. Synthesis and applications: Our results confirm that increasing early‐season mass‐flowering crop cover dilutes wild pollinators in crop habitats during bloom. We find that dilution effects were absent late in the season. While mass‐flowering crop cultivation alone is unlikely to be sufficient for maintaining pollinators, as part of carefully designed diverse crop rotations or mixtures combined with the preservation of permanent non‐crop habitats, it might provide valuable supplementary food resources for pollinators in temperate agroecosystems, particularly later in the season when alternative flower resources are scarce