Regularity for eigenfunctions of Schr\"odinger operators

We prove a regularity result in weighted Sobolev spaces (or Babuska--Kondratiev spaces) for the eigenfunctions of a Schr\"odinger operator. More precisely, let K_{a}^{m}(\mathbb{R}^{3N}) be the weighted Sobolev space obtained by blowing up the set of singular points of the Coulomb type potential V(x) = \sum_{1 \le j \le N} \frac{b_j}{|x_j|} + \sum_{1 \le i < j \le N} \frac{c_{ij}}{|x_i-x_j|}, x in \mathbb{R}^{3N}, b_j, c_{ij} in \mathbb{R}. If u in L^2(\mathbb{R}^{3N}) satisfies (-\Delta + V) u = \lambda u in distribution sense, then u belongs to K_{a}^{m} for all m \in \mathbb{Z}_+ and all a \le 0. Our result extends to the case when b_j and c_{ij} are suitable bounded functions on the blown-up space. In the single-electron, multi-nuclei case, we obtain the same result for all a<3/2.Comment: to appear in Lett. Math. Phy

On the analogy between streamlined magnetic and solid obstacles

Analogies are elaborated in the qualitative description of two systems: the magnetohydrodynamic (MHD) flow moving through a region where an external local magnetic field (magnetic obstacle) is applied, and the ordinary hydrodynamic flow around a solid obstacle. The former problem is of interest both practically and theoretically, and the latter one is a classical problem being well understood in ordinary hydrodynamics. The first analogy is the formation in the MHD flow of an impenetrable region -- core of the magnetic obstacle -- as the interaction parameter $N$, i.e. strength of the applied magnetic field, increases significantly. The core of the magnetic obstacle is streamlined both by the upstream flow and by the induced cross stream electric currents, like a foreign insulated insertion placed inside the ordinary hydrodynamic flow. In the core, closed streamlines of the mass flow resemble contour lines of electric potential, while closed streamlines of the electric current resemble contour lines of pressure. The second analogy is the breaking away of attached vortices from the recirculation pattern produced by the magnetic obstacle when the Reynolds number $Re$, i.e. velocity of the upstream flow, is larger than a critical value. This breaking away of vortices from the magnetic obstacle is similar to that occurring past a real solid obstacle. Depending on the inlet and/or initial conditions, the observed vortex shedding can be either symmetric or asymmetric.Comment: minor changes, accepted for PoF, 26 pages, 7 figure

An introduction to the Australian and New Zealand flux tower network - OzFlux

Published: 31 October 2016OzFlux is the regional Australian and New Zealand flux tower network that aims to provide a continental-scale national research facility to monitor and assess trends, and improve predictions, of Australia's terrestrial biosphere and climate. This paper describes the evolution, design, and current status of OzFlux as well as provides an overview of data processing. We analyse measurements from all sites within the Australian portion of the OzFlux network and two sites from New Zealand. The response of the Australian biomes to climate was largely consistent with global studies except that Australian systems had a lower ecosystem water-use efficiency. Australian semi-arid/arid ecosystems are important because of their huge extent (70 %) and they have evolved with common moisture limitations. We also found that Australian ecosystems had a similar radiation-use efficiency per unit leaf area compared to global values that indicates a convergence toward a similar biochemical efficiency. The two New Zealand sites represented extremes in productivity for a moist temperate climate zone, with the grazed dairy farm site having the highest GPP of any OzFlux site (2620 gC m⁻² yr⁻¹) and the natural raised peat bog site having a very low GPP (820 gC m⁻² yr⁻¹). The paper discusses the utility of the flux data and the synergies between flux, remote sensing, and modelling. Lastly, the paper looks ahead at the future direction of the network and concludes that there has been a substantial contribution by OzFlux, and considerable opportunities remain to further advance our understanding of ecosystem response to disturbances, including drought, fire, land-use and land-cover change, land management, and climate change, which are relevant both nationally and internationally. It is suggested that a synergistic approach is required to address all of the spatial, ecological, human, and cultural challenges of managing the delicately balanced ecosystems in Australasia.Jason Beringer ... Wayne Meyer ... et al

Numerical Computations with H(div)-Finite Elements for the Brinkman Problem

The H(div)-conforming approach for the Brinkman equation is studied numerically, verifying the theoretical a priori and a posteriori analysis in previous work of the authors. Furthermore, the results are extended to cover a non-constant permeability. A hybridization technique for the problem is presented, complete with a convergence analysis and numerical verification. Finally, the numerical convergence studies are complemented with numerical examples of applications to domain decomposition and adaptive mesh refinement.Comment: Minor clarifications, added references. Reordering of some figures. To appear in Computational Geosciences, final article available at http://www.springerlink.co

Tips for implementing multigrid methods on domains containing holes

As part of our development of a computer code to perform 3D `constrained evolution' of Einstein's equations in 3+1 form, we discuss issues regarding the efficient solution of elliptic equations on domains containing holes (i.e., excised regions), via the multigrid method. We consider as a test case the Poisson equation with a nonlinear term added, as a means of illustrating the principles involved, and move to a "real world" 3-dimensional problem which is the solution of the conformally flat Hamiltonian constraint with Dirichlet and Robin boundary conditions. Using our vertex-centered multigrid code, we demonstrate globally second-order-accurate solutions of elliptic equations over domains containing holes, in two and three spatial dimensions. Keys to the success of this method are the choice of the restriction operator near the holes and definition of the location of the inner boundary. In some cases (e.g. two holes in two dimensions), more and more smoothing may be required as the mesh spacing decreases to zero; however for the resolutions currently of interest to many numerical relativists, it is feasible to maintain second order convergence by concentrating smoothing (spatially) where it is needed most. This paper, and our publicly available source code, are intended to serve as semi-pedagogical guides for those who may wish to implement similar schemes.Comment: 18 pages, 11 figures, LaTeX. Added clarifications and references re. scope of paper, mathematical foundations, relevance of work. Accepted for publication in Classical & Quantum Gravit

Scalability of Incompressible Flow Computations on Multi-GPU Clusters Using Dual-Level and Tri-Level Parallelism

High performance computing using graphics processing units (GPUs) is gaining popularity in the scientific computing field, with many large compute clusters being augmented with multiple GPUs in each node. We investigate hybrid tri-level (MPI-OpenMP-CUDA) parallel implementations to explore the efficiency and scalability of incompressible flow computations on GPU clusters up to 128 GPUS. This work details some of the unique issues faced when merging fine-grain parallelism on the GPU using CUDA with coarse-grain parallelism using OpenMP for intra-node and MPI for inter-node communication. Comparisons between the tri-level MPI-OpenMP-CUDA and dual-level MPI-CUDA implementations are shown using computationally large computational fluid dynamics (CFD) simulations. Our results demonstrate that a tri-level parallel implementation does not provide a significant advantage in performance over the dual-level implementation, however further research is needed to justify our conclusion for a cluster with a high GPU per node density or when using software that can utilize OpenMP’s fine-grain parallelism more effectively

Carbon uptake and water use in woodlands and forests in southern Australia during an extreme heat wave event in the ‘Angry Summer’ of 2012/2013

As a result of climate change warmer temperatures are projected through the 21st century and are already increasing above modelled predictions. Apart from increases in the mean, warm/hot temperature extremes are expected to become more prevalent in the future, along with an increase in the frequency of droughts. It is crucial to better understand the response of terrestrial ecosystems to such temperature extremes for predicting land-surface feedbacks in a changing climate. While land-surface feedbacks in drought conditions and during heat waves have been reported from Europe and the US, direct observations of the impact of such extremes on the carbon and water cycles in Australia have been lacking. During the 2012/2013 summer, Australia experienced a record-breaking heat wave with an exceptional spatial extent that lasted for several weeks. In this study we synthesised eddy-covariance measurements from seven woodlands and one forest site across three biogeographic regions in southern Australia. These observations were combined with model results from BIOS2 (Haverd et al., 2013a, b) to investigate the effect of the summer heat wave on the carbon and water exchange of terrestrial ecosystems which are known for their resilience toward hot and dry conditions. We found that water-limited woodland and energy-limited forest ecosystems responded differently to the heat wave. During the most intense part of the heat wave, the woodlands experienced decreased latent heat flux (23 % of background value), increased Bowen ratio (154 %) and reduced carbon uptake (60 %). At the same time the forest ecosystem showed increased latent heat flux (151 %), reduced Bowen ratio (19 %) and increased carbon uptake (112 %). Higher temperatures caused increased ecosystem respiration at all sites (up to 139 %). During daytime all ecosystems remained carbon sinks, but carbon uptake was reduced in magnitude. The number of hours during which the ecosystem acted as a carbon sink was also reduced, which switched the woodlands into a carbon source on a daily average. Precipitation occurred after the first, most intense part of the heat wave, and the subsequent cooler temperatures in the temperate woodlands led to recovery of the carbon sink, decreased the Bowen ratio (65 %) and hence increased evaporative cooling. Gross primary productivity in the woodlands recovered quickly with precipitation and cooler temperatures but respiration remained high. While the forest proved relatively resilient to this short-term heat extreme the response of the woodlands is the first direct evidence that the carbon sinks of large areas of Australia may not be sustainable in a future climate with an increased number, intensity and duration of heat waves.Eva van Gorsel, Sebastian Wolf, James Cleverly, Peter Isaac, Vanessa Haverd, Cäcilia Ewenz, Stefan Arndt, Jason Beringer, Víctor Resco de Dios, Bradley J. Evans, Anne Griebel, Lindsay B. Hutley, Trevor Keenan, Natascha Kljun, Craig Macfarlane, Wayne S. Meyer, Ian McHugh, Elise Pendall, Suzanne M. Prober and Richard Silberstei

Towards a unified theory of Sobolev inequalities

We discuss our work on pointwise inequalities for the gradient which are connected with the isoperimetric profile associated to a given geometry. We show how they can be used to unify certain aspects of the theory of Sobolev inequalities. In particular, we discuss our recent papers on fractional order inequalities, Coulhon type inequalities, transference and dimensionless inequalities and our forthcoming work on sharp higher order Sobolev inequalities that can be obtained by iteration.Comment: 39 pages, made some changes to section 1

An introduction to the Australian and New Zealand flux tower network - OzFlux

© Author(s) 2016. OzFlux is the regional Australian and New Zealand flux tower network that aims to provide a continental-scale national research facility to monitor and assess trends, and improve predictions, of Australia's terrestrial biosphere and climate. This paper describes the evolution, design, and current status of OzFlux as well as provides an overview of data processing. We analyse measurements from all sites within the Australian portion of the OzFlux network and two sites from New Zealand. The response of the Australian biomes to climate was largely consistent with global studies except that Australian systems had a lower ecosystem water-use efficiency. Australian semi-arid/arid ecosystems are important because of their huge extent (70 %) and they have evolved with common moisture limitations. We also found that Australian ecosystems had a similar radiation-use efficiency per unit leaf area compared to global values that indicates a convergence toward a similar biochemical efficiency. The two New Zealand sites represented extremes in productivity for a moist temperate climate zone, with the grazed dairy farm site having the highest GPP of any OzFlux site (2620 gC m-2 yr-1) and the natural raised peat bog site having a very low GPP (820 gC m-2 yr-1). The paper discusses the utility of the flux data and the synergies between flux, remote sensing, and modelling. Lastly, the paper looks ahead at the future direction of the network and concludes that there has been a substantial contribution by OzFlux, and considerable opportunities remain to further advance our understanding of ecosystem response to disturbances, including drought, fire, land-use and land-cover change, land management, and climate change, which are relevant both nationally and internationally. It is suggested that a synergistic approach is required to address all of the spatial, ecological, human, and cultural challenges of managing the delicately balanced ecosystems in Australasia