Gradient recovery in adaptive finite-element methods for parabolic problems

We derive energy-norm a posteriori error bounds, using gradient recovery (ZZ) estimators to control the spatial error, for fully discrete schemes for the linear heat equation. This appears to be the �rst completely rigorous derivation of ZZ estimators for fully discrete schemes for evolution problems, without any restrictive assumption on the timestep size. An essential tool for the analysis is the elliptic reconstruction technique.Our theoretical results are backed with extensive numerical experimentation aimed at (a) testing the practical sharpness and asymptotic behaviour of the error estimator against the error, and (b) deriving an adaptive method based on our estimators. An extra novelty provided is an implementation of a coarsening error "preindicator", with a complete implementation guide in ALBERTA in the appendix.

Effect of Rht alleles on the tolerance of wheat grain set to high temperature and drought stress during booting and anthesis

Factorial pot experiments were conducted to compare the responses of GA-sensitive and GA-insensitive reduced height (Rht) alleles in wheat for susceptibility to heat and drought stress during booting and anthesis. Grain set (grains/spikelet) of near isogenic lines (NILs) was assessed following three day transfers to controlled environments imposing day temperatures (t) from 20 to 40°C. Transfers were during booting and/or anthesis and pots maintained at field capacity (FC) or had water withheld. Logistic responses (y = c/1+e-b(t -m)) described declining grain set with increasing t, and t5 was that fitted to give a 5% reduction in grain set. Averaged over NIL, t5 for anthesis at FC was 31.7±0.47°C (S.E.M, 26 d.f.). Drought at anthesis reduced t5 by <2°C. Maintaining FC at booting conferred considerable resistance to high temperatures (t5=33.9°C) but booting was particularly heat susceptible without water (t5 =26.5°C). In one background (cv. Mercia), for NILs varying at the Rht-D1 locus, there was progressive reduction in t5 with dwarfing and reduced gibberellic acid (GA) sensitivity (Rht-D1a, tall, 32.7±0.72; Rht-D1b, semi-dwarf, 29.5±0.85; Rht-D1c, severe dwarf, 24.2±0.72). This trend was not evident for the Rht-B1 locus, or for Rht-D1b in an alternative background (Maris Widgeon). The GA-sensitive severe dwarf Rht12 was more heat tolerant (t5=29.4±0.72) than the similarly statured GA-insensitive Rht-D1c. The GA-sensitive, semi-dwarfing Rht8 conferred greater drought tolerance in one experiment. Despite the effects of Rht-D1 alleles in Mercia on stress tolerance, the inconsistency of the effects over background and locus led to the conclusion that semi-dwarfing with GA-insensitivity did not necessarily increase sensitivity to stress at booting and flowering. In comparison to effects of semi-dwarfing alleles, responses to heat stress are much more dramatically affected by water availability and the precise growth stage at which the stress is experienced by the plants.

Is the Helmholtz equation really sign-indefinite?

The usual variational (or weak) formulations of the Helmholtz equation are sign-indefinite in the sense that the bilinear forms cannot be bounded below by a positive multiple of the appropriate norm squared. This is often for a good reason, since in bounded domains under certain boundary conditions the solution of the Helmholtz equation is not unique at wavenumbers that correspond to eigenvalues of the Laplacian, and thus the variational problem cannot be sign-definite. However, even in cases where the solution is unique for all wavenumbers, the standard variational formulations of the Helmholtz equation are still indefinite when the wavenumber is large. This indefiniteness has implications for both the analysis and the practical implementation of finite element methods. In this paper we introduce new sign-definite (also called coercive or elliptic) formulations of the Helmholtz equation posed in either the interior of a star-shaped domain with impedance boundary conditions, or the exterior of a star-shaped domain with Dirichlet boundary conditions. Like the standard variational formulations, these new formulations arise just by multiplying the Helmholtz equation by particular test functions and integrating by parts.

Fireball models for flares in AE Aquarii

We examine the flaring behaviour of the cataclysmic variable AE Aqr in the context of the `magnetic propeller' model for this system. The flares are thought to arise from collisions between high-density regions in the material expelled from the system after interaction with the rapidly rotating magnetosphere of the white dwarf. We calculate the first quantitative models for the flaring and calculate the time-dependent emergent optical spectra from the resulting hot, expanding ball of gas. We compare the results under different assumptions to observations and derive values for the mass, length-scale and temperature of the material involved in the flare. We see that the fits suggest that the secondary star in this system has Population II composition.

High-speed Keck spectroscopy of flares and oscillations in AE Aquarii

Using high-time-resolution (72 ms) spectroscopy of AE Aqr obtained with LRIS on Keck II we have determined the spectrum and spectral evolution of a small flare. Continuum and integrated line fluxes in the flare spectrum are measured, and the evolution of the flare is parametrized for future comparison with detailed models of the flares. We find that the velocities of the flaring components are consistent with those previously reported for AE Aqr by Welsh, Horne & Gomer and Horne. The characteristics of the 33-s oscillations are investigated: we derive the oscillation amplitude spectrum, and from that determine the spectrum of the heated regions on the rotating white dwarf. Blackbody fits to the major and minor pulse spectra and an analysis of the emission-line oscillation properties highlight the shortfalls in the simple hotspot model for the oscillations.

Rapid Keck Spectroscopy of Cataclysmic Variables

We present an analysis of Rapid Keck Spectroscopy of the CVs AM Her (polar) and SS Cyg (dwarf nova). We decompose the spectra into constant and variable components and identify different types of variability in AM Her with different characteristic timescales. The variable flickering component of the accretion disc flux and the observational characteristics of a small flare in SS Cyg are isolated.

REVIEW: Fireballs, Flares and Flickering

We review our understanding of the prototype ``Propeller'' system AE Aqr and we examine its flaring behaviour in detail. The flares are thought to arise from collisions between high density regions in the material expelled from the system after interaction with the rapidly rotating magnetosphere of the white dwarf. We show calculations of the time-dependent emergent optical spectra from the resulting hot, expanding ball of gas and derive values for the mass, lengthscale and temperature of the material involved. We see that the fits suggest that the secondary star in this system has reduced metal abundances and that, counter-intuitively, the evolution of the fireballs is best modelled as isothermal.

Continuum sea ice rheology determined from subcontinuum mechanics

[1] A method is presented to calculate the continuum-scale sea ice stress as an imposed, continuum-scale strain-rate is varied. The continuum-scale stress is calculated as the area-average of the stresses within the floes and leads in a region (the continuum element). The continuum-scale stress depends upon: the imposed strain rate; the subcontinuum scale, material rheology of sea ice; the chosen configuration of sea ice floes and leads; and a prescribed rule for determining the motion of the floes in response to the continuum-scale strain-rate. We calculated plastic yield curves and flow rules associated with subcontinuum scale, material sea ice rheologies with elliptic, linear and modified Coulombic elliptic plastic yield curves, and with square, diamond and irregular, convex polygon-shaped floes. For the case of a tiling of square floes, only for particular orientations of the leads have the principal axes of strain rate and calculated continuum-scale sea ice stress aligned, and these have been investigated analytically. The ensemble average of calculated sea ice stress for square floes with uniform orientation with respect to the principal axes of strain rate yielded alignment of average stress and strain-rate principal axes and an isotropic, continuum-scale sea ice rheology. We present a lemon-shaped yield curve with normal flow rule, derived from ensemble averages of sea ice stress, suitable for direct inclusion into the current generation of sea ice models. This continuum-scale sea ice rheology directly relates the size (strength) of the continuum-scale yield curve to the material compressive strength.

A comparison of duality and energy a posteriori estimates for $\mathrm {L}_{\infty }(0,T;\mathrm {L}_2(\varOmega ))$ in parabolic problems

We use the elliptic reconstruction technique in combination with a duality approach to prove a posteriori error estimates for fully discrete backward Euler scheme for linear parabolic equations. As an application, we combine our result with the residual based estimators from the a posteriori estimation for elliptic problems to derive space-error indicators and thus a fully practical version of the estimators bounding the error in the $ \mathrm {L}_{\infty }(0,T;\mathrm {L}_2(\varOmega ))$ norm. These estimators, which are of optimal order, extend those introduced by Eriksson and Johnson in 1991 by taking into account the error induced by the mesh changes and allowing for a more flexible use of the elliptic estimators. For comparison with previous results we derive also an energy-based a posteriori estimate for the $ \mathrm {L}_{\infty }(0,T;\mathrm {L}_2(\varOmega ))$-error which simplifies a previous one given by Lakkis and Makridakis in 2006. We then compare both estimators (duality vs. energy) in practical situations and draw conclusions.

Reduced relative entropy techniques for a posteriori analysis of multiphase problems in elastodynamics

We give an a posteriori analysis of a semidiscrete discontinuous Galerkin scheme approximating solutions to a model of multiphase elastodynamics, which involves an energy density depending not only on the strain but also the strain gradient. A key component in the analysis is the reduced relative entropy stability framework developed in Giesselmann (2014, SIAM J. Math. Anal., 46, 3518–3539). This framework allows energy-type arguments to be applied to continuous functions. Since we advocate the use of discontinuous Galerkin methods we make use of two families of reconstructions, one set of discrete reconstructions and a set of elliptic reconstructions to apply the reduced relative entropy framework in this setting.

On the structure of infinity-harmonic maps

Let H ∈ C 2(ℝ N×n ), H ≥ 0. The PDE system arises as the Euler-Lagrange PDE of vectorial variational problems for the functional E ∞(u, Ω) = ‖H(Du)‖ L ∞(Ω) defined on maps u: Ω ⊆ ℝ n → ℝ N . (1) first appeared in the author's recent work. The scalar case though has a long history initiated by Aronsson. Herein we study the solutions of (1) with emphasis on the case of n = 2 ≤ N with H the Euclidean norm on ℝ N×n , which we call the “∞-Laplacian”. By establishing a rigidity theorem for rank-one maps of independent interest, we analyse a phenomenon of separation of the solutions to phases with qualitatively different behaviour. As a corollary, we extend to N ≥ 2 the Aronsson-Evans-Yu theorem regarding non existence of zeros of |Du| and prove a maximum principle. We further characterise all H for which (1) is elliptic and also study the initial value problem for the ODE system arising for n = 1 but with H(·, u, u′) depending on all the arguments.

Noether-type discrete conserved quantities arising from a finite element approximation of a variational problem

In this work, we prove a weak Noether-type Theorem for a class of variational problems that admit broken extremals. We use this result to prove discrete Noether-type conservation laws for a conforming finite element discretisation of a model elliptic problem. In addition, we study how well the finite element scheme satisfies the continuous conservation laws arising from the application of Noether’s first theorem (1918). We summarise extensive numerical tests, illustrating the conservation of the discrete Noether law using the p-Laplacian as an example and derive a geometric-based adaptive algorithm where an appropriate Noether quantity is the goal functional.

A new three-locus model for rootstock-induced dwarfing in apple revealed by genetic mapping of root bark percentage

Rootstock-induced dwarfing of apple scions revolutionized global apple production during the twentieth century, leading to the development of modern intensive orchards. A high root bark percentage (the percentage of the whole root area constituted by root cortex) has previously been associated with rootstock induced dwarfing in apple. In this study, the root bark percentage was measured in a full-sib family of ungrafted apple rootstocks and found to be under the control of three loci. Two QTL for root bark percentage were found to co-localise to the same genomic regions on chromosome 5 and chromosome 11 previously identified as controlling dwarfing, Dw1 and Dw2, respectively. A third QTL was identified on chromosome 13 in a region that has not been previously associated with dwarfing. The development of closely linked 3 Sequence-tagged site STS markers improved the resolution of allelic classes thereby allowing the detection of dominance and epistatic interactions between loci, with high root bark percentage only occurring in specific allelic combinations. In addition, we report a significant negative correlation between root bark percentage and stem diameter (an indicator of tree vigour), measured on a clonally propagated grafted subset of the mapping population. The demonstrated link between root bark percentage and rootstock-induced dwarfing of the scion leads us to propose a three-locus model that is able to explain levels of dwarfing from the dwarf ‘M.27’ to the semi-invigorating rootstock ‘M.116’. Moreover, we suggest that the QTL on chromosome 13 (Rb3) might be analogous to a third dwarfing QTL, Dw3 that has not previously been identified.