Macro-element interpolation on tensor product meshes

A general theory for obtaining anisotropic interpolation error estimates for macro-element interpolation is developed revealing general construction principles. We apply this theory to interpolation operators on a macro type of biquadratic $C^1$ finite elements on rectangle grids which can be viewed as a rectangular version of the $C^1$ Powell-Sabin element. This theory also shows how interpolation on the Bogner-Fox-Schmidt finite element space (or higher order generalizations) can be analyzed in a unified framework. Moreover we discuss a modification of Scott-Zhang type giving optimal error estimates under the regularity required without imposing quasi uniformity on the family of macro-element meshes used. We introduce and analyze an anisotropic macro-element interpolation operator, which is the tensor product of one-dimensional $C^1-P_2$ macro interpolation and $P_2$ Lagrange interpolation. These results are used to approximate the solution of a singularly perturbed reaction-diffusion problem on a Shishkin mesh that features highly anisotropic elements. Hereby we obtain an approximation whose normal derivative is continuous along certain edges of the mesh, enabling a more sophisticated analysis of a continuous interior penalty method in another paper

Error analysis of the Galerkin FEM in L 2 -based norms for problems with layers: On the importance, conception and realization of balancing

In the present thesis it is shown that the most natural choice for a norm for the analysis of the Galerkin FEM, namely the energy norm, fails to capture the boundary layer functions arising in certain reaction-diffusion problems. In view of a formal Definition such reaction-diffusion problems are not singularly perturbed with respect to the energy norm. This observation raises two questions: 1. Does the Galerkin finite element method on standard meshes yield satisfactory approximations for the reaction-diffusion problem with respect to the energy norm? 2. Is it possible to strengthen the energy norm in such a way that the boundary layers are captured and that it can be reconciled with a robust finite element method, i.e.~robust with respect to this strong norm? In Chapter 2 we answer the first question. We show that the Galerkin finite element approximation converges uniformly in the energy norm to the solution of the reaction-diffusion problem on standard shape regular meshes. These results are completely new in two dimensions and are confirmed by numerical experiments. We also study certain convection-diffusion problems with characterisitc layers in which some layers are not well represented in the energy norm. These theoretical findings, validated by numerical experiments, have interesting implications for adaptive methods. Moreover, they lead to a re-evaluation of other results and methods in the literature. In 2011 Lin and Stynes were the first to devise a method for a reaction-diffusion problem posed in the unit square allowing for uniform a priori error estimates in an adequate so-called balanced norm. Thus, the aforementioned second question is answered in the affirmative. Obtaining a non-standard weak formulation by testing also with derivatives of the test function is the key idea which is related to the H^1-Galerkin methods developed in the early 70s. Unfortunately, this direct approach requires excessive smoothness of the finite element space considered. Lin and Stynes circumvent this problem by rewriting their problem into a first order system and applying a mixed method. Now the norm captures the layers. Therefore, they need to be resolved by some layer-adapted mesh. Lin and Stynes obtain optimal error estimates with respect to the balanced norm on Shishkin meshes. However, their method is unable to preserve the symmetry of the problem and they rely on the Raviart-Thomas element for H^div-conformity. In Chapter 4 of the thesis a new continuous interior penalty (CIP) method is present, embracing the approach of Lin and Stynes in the context of a broken Sobolev space. The resulting method induces a balanced norm in which uniform error estimates are proven. In contrast to the mixed method the CIP method uses standard Q_2-elements on the Shishkin meshes. Both methods feature improved stability properties in comparison with the Galerkin FEM. Nevertheless, the latter also yields approximations which can be shown to converge to the true solution in a balanced norm uniformly with respect to diffusion parameter. Again, numerical experiments are conducted that agree with the theoretical findings. In every finite element analysis the approximation error comes into play, eventually. If one seeks to prove any of the results mentioned on an anisotropic family of Shishkin meshes, one will need to take advantage of the different element sizes close to the boundary. While these are ideally suited to reflect the solution behavior, the error analysis is more involved and depends on anisotropic interpolation error estimates. In Chapter 3 the beautiful theory of Apel and Dobrowolski is extended in order to obtain anisotropic interpolation error estimates for macro-element interpolation. This also sheds light on fundamental construction principles for such operators. The thesis introduces a non-standard finite element space that consists of biquadratic C^1-finite elements on macro-elements over tensor product grids, which can be viewed as a rectangular version of the C^1-Powell-Sabin element. As an application of the general theory developed, several interpolation operators mapping into this FE space are analyzed. The insight gained can also be used to prove anisotropic error estimates for the interpolation operator induced by the well-known C^1-Bogner-Fox-Schmidt element. A special modification of Scott-Zhang type and a certain anisotropic interpolation operator are also discussed in detail. The results of this chapter are used to approximate the solution to a recation-diffusion-problem on a Shishkin mesh that features highly anisotropic elements. The obtained approximation features continuous normal derivatives across certain edges of the mesh, enabling the analysis of the aforementioned CIP method.:Notation 1 Introduction 2 Galerkin FEM error estimation in weak norms 2.1 Reaction-diffusion problems 2.2 A convection-diffusion problem with weak characteristic layers and a Neumann outflow condition 2.3 A mesh that resolves only part of the exponential layer and neglects the weaker characteristic layers 2.3.1 Weakly imposed characteristic boundary conditions 2.4 Numerical experiments 2.4.1 A reaction-diffusion problem with boundary layers 2.4.2 A reaction-diffusion problem with an interior layer 2.4.3 A convection-diffusion problem with characteristic layers and a Neumann outflow condition 2.4.4 A mesh that resolves only part of the exponential layer and neglects the weaker characteristic layers 3 Macro-interpolation on tensor product meshes 3.1 Introduction 3.2 Univariate C1-P2 macro-element interpolation 3.3 C1-Q2 macro-element interpolation on tensor product meshes 3.4 A theory on anisotropic macro-element interpolation 3.5 C1 macro-interpolation on anisotropic tensor product meshes 3.5.1 A reduced macro-element interpolation operator 3.5.2 The full C1-Q2 interpolation operator 3.5.3 A C1-Q2 macro-element quasi-interpolation operator of Scott-Zhang type on tensor product meshes 3.5.4 Summary: anisotropic C1 (quasi-)interpolation error estimates 3.6 An anisotropic macro-element of tensor product type 3.7 Application of macro-element interpolation on a tensor product Shishkin mesh 4 Balanced norm results for reaction-diffusion 4.1 The balanced finite element method of Lin and Stynes 4.2 A C0 interior penalty method 4.3 Galerkin finite element method 4.3.1 L2-norm error bounds and supercloseness 4.3.2 Maximum-norm error bounds 4.4 Numerical verification 4.5 Further developments and summary Reference

Quality of life and surgical outcome of ABBA versus EndoCATS endoscopic thyroid surgery: a single center experience

BACKGROUND Thyroid surgery is often performed, especially in young female patients. As patient satisfaction become more and more important, different extra-cervical \textquotedblremote\textquotedbl approaches have evolved to avoid visible scars in the neck for better cosmetic outcome. The most common remote approaches are the transaxillary and retroauricular. Aim of this work is to compare Endoscopic Cephalic Access Thyroid Surgery (EndoCATS) and axillo-bilateral-breast approach (ABBA) to standard open procedures regarding perioperative outcome and in addition to control cohorts regarding quality of life (QoL) and patient satisfaction. METHODS In a single center, 59 EndoCATS und 52 ABBA procedures were included out of a 2 years period and compared to 225 open procedures using propensity-score matching. For the endoscopic procedures, cosmetic outcome, patient satisfaction and QoL (SF-12 questionnaire) were examined in prospective follow-up. For QoL a German standard cohort and non-surgically patients with thyroid disease were used as controls. RESULT The overall perioperative outcome was similar for all endoscopic compared to open thyroid surgeries. Surgical time was longer for endoscopic procedures. There were no cases of permanent hypoparathyroidism and no significant differences regarding temporary or permanent recurrent laryngeal nerve (RLN) palsies between open and ABBA or EndoCATS procedures (χ2; p = 0.893 and 0.840). For ABBA and EndoCATS, 89.6% and 94.2% of patients were satisfied with the surgical procedure. Regarding QoL, there was an overall significant difference in distribution for physical, but not for mental health between groups (p < 0.001 and 0.658). Both endoscopic groups performed slightly worse regarding physical health, but without significant difference between the individual groups in post hoc multiple comparison. CONCLUSION Endoscopic thyroid surgery is safe with comparable perioperative outcome in experienced high-volume centers. Patient satisfaction and cosmetic results are excellent; QoL is impaired in surgical patients, as they perform slightly worse compared to German standard cohort and non-surgical patients

Influence of the barotropic mean flow on the width and the structure of the Atlantic Equatorial Deep Jets

A representation of an equatorial basin mode excited in a shallow water model for a single high order baroclinic vertical normal mode is used as a simple model for the equatorial deep jets. The model is linearized about both a state of rest and a barotropic mean flow corresponding to the observed Atlantic Equatorial Intermediate Current System. We found that the eastward mean flow associated with the North and South Intermediate Counter Currents (NICC and SICC, respectively) effectively shields the Equator from off-equatorial Rossby waves. The westward propagation of these waves is blocked and focusing on the Equator due to beta dispersion is prevented. This leads to less energetic jets along the Equator. On the other hand, the westward barotropic mean flow along the Equator reduces the gradient of absolute vorticity and hence widens the cross-equatorial structure of the basin mode. Increasing lateral viscosity predominantly affects the width of the basin modes’ Kelvin wave component in the presence of the mean flow while the Rossby wave is confined by the flanking NICC and SICC. Independent of the presence of the mean flow, the application of sufficient lateral mixing also hinders the focusing of off-equatorial Rossby waves, which is hence an unlikely feature of a low-frequency basin mode in the real ocean

Origin of Life

The evolution of life has been a big enigma despite rapid advancements in the fields of biochemistry, astrobiology, and astrophysics in recent years. The answer to this puzzle has been as mind-boggling as the riddle relating to evolution of Universe itself. Despite the fact that panspermia has gained considerable support as a viable explanation for origin of life on the Earth and elsewhere in the Universe, the issue remains far from a tangible solution. This paper examines the various prevailing hypotheses regarding origin of life like abiogenesis, RNA World, Iron-sulphur World, and panspermia; and concludes that delivery of life-bearing organic molecules by the comets in the early epoch of the Earth alone possibly was not responsible for kick-starting the process of evolution of life on our planet.Comment: 32 pages, 8 figures,invited review article, minor additio

Calcitization of aragonitic bryozoans in Cenozoic tropical carbonates from East Kalimantan, Indonesia

© The Author(s) 2016. Open Access. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The file attached is the published version of the article

A morphogram for silica-witherite biomorphs and its application to microfossil identification in the early earth rock record

Archean hydrothermal environments formed a likely site for the origin and early evolution of life. These are also the settings, however, were complex abiologic structures can form. Low-temperature serpentinization of ultramafic crust can generate alkaline, silica-saturated fluids in which carbonate–silica crystalline aggregates with life-like morphologies can self-assemble. These “biomorphs” could have adsorbed hydrocarbons from Fischer–Tropsch type synthesis processes, leading to metamorphosed structures that resemble carbonaceous microfossils. Although this abiogenic process has been extensively cited in the literature and has generated important controversy, so far only one specific biomorph type with a filamentous shape has been discussed for the interpretation of Archean microfossils. It is therefore critical to precisely determine the full distribution in morphology and size of these biomorphs, and to study the range of plausible geochemical conditions under which these microstructures can form. Here, a set of witherite-silica biomorph synthesis experiments in silica-saturated solutions is presented, for a range of pH values (from 9 to 11.5) and barium ion concentrations (from 0.6 to 40 mmol/L BaCl2). Under these varying conditions, a wide range of life-like structures is found, from fractal dendrites to complex shapes with continuous curvature. The size, spatial concentration, and morphology of the biomorphs are strongly controlled by environmental parameters, among which pH is the most important. This potentially limits the diversity of environments in which the growth of biomorphs could have occurred on Early Earth. Given the variety of the observed biomorph morphologies, our results show that the morphology of an individual microstructure is a poor criterion for biogenicity. However, biomorphs may be distinguished from actual populations of cellular microfossils by their wide, unimodal size distribution. Biomorphs grown by diffusion in silica gel can be differentiated by their continuous gradient in size, spatial density, and morphology along the direction of diffusion.This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement nº 646894) and under the ERC Seventh Framework Programme FP7/2007-2013 (grant agreement n° 340863). JMG-R also acknowledges the Ministerio de Economía y Competitividad of Spain through the project CGL2016-78971-P. We acknowledge the analytical platform PARI and Stefan Borenstazjn for SEM imaging. Prof. Y. Tsukii and The Protist Information Server (http://protist.i.hosei.ac.jp/) are acknowledged for the use of pictures of cyanobacteria. This is IPGP contribution n° 3912. We are grateful to two anonymous reviewers for their helpful comments

The faint young Sun problem

For more than four decades, scientists have been trying to find an answer to one of the most fundamental questions in paleoclimatology, the `faint young Sun problem'. For the early Earth, models of stellar evolution predict a solar energy input to the climate system which is about 25% lower than today. This would result in a completely frozen world over the first two billion years in the history of our planet, if all other parameters controlling Earth's climate had been the same. Yet there is ample evidence for the presence of liquid surface water and even life in the Archean (3.8 to 2.5 billion years before present), so some effect (or effects) must have been compensating for the faint young Sun. A wide range of possible solutions have been suggested and explored during the last four decades, with most studies focusing on higher concentrations of atmospheric greenhouse gases like carbon dioxide, methane or ammonia. All of these solutions present considerable difficulties, however, so the faint young Sun problem cannot be regarded as solved. Here I review research on the subject, including the latest suggestions for solutions of the faint young Sun problem and recent geochemical constraints on the composition of Earth's early atmosphere. Furthermore, I will outline the most promising directions for future research. In particular I would argue that both improved geochemical constraints on the state of the Archean climate system and numerical experiments with state-of-the-art climate models are required to finally assess what kept the oceans on the Archean Earth from freezing over completely.Comment: 32 pages, 8 figures. Invited review paper accepted for publication in Reviews of Geophysic