An Exponentially Convergent Nonpolynomial Finite Element Method for Time-Harmonic Scattering from Polygons

In recent years nonpolynomial finite element methods have received increasing attention for the efficient solution of wave problems. As with their close cousin the method of particular solutions, high efficiency comes from using solutions to the Helmholtz equation as basis functions. We present and analyze such a method for the scattering of two-dimensional scalar waves from a polygonal domain that achieves exponential convergence purely by increasing the number of basis functions in each element. Key ingredients are the use of basis functions that capture the singularities at corners and the representation of the scattered field towards infinity by a combination of fundamental solutions. The solution is obtained by minimizing a least-squares functional, which we discretize in such a way that a matrix least-squares problem is obtained. We give computable exponential bounds on the rate of convergence of the least-squares functional that are in very good agreement with the observed numerical convergence. Challenging numerical examples, including a nonconvex polygon with several corner singularities, and a cavity domain, are solved to around 10 digits of accuracy with a few seconds of CPU time. The examples are implemented concisely with MPSpack, a MATLAB toolbox for wave computations with nonpolynomial basis functions, developed by the authors. A code example is included.

Switching on a two-dimensional time-harmonic scalar wave in the presence of a diffracting edge

This paper concerns the switching on of two-dimensional time-harmonic scalar waves. We first review the switch-on problem for a point source in free space, then proceed to analyse the analogous problem for the diffraction of a plane wave by a half-line (the ‘Sommerfeld problem’), determining in both cases the conditions under which the field is well-approximated by the solution of the corresponding frequency domain problem. In both cases the rate of convergence to the frequency domain solution is found to be dependent on the strength of the singularity on the leading wavefront. In the case of plane wave diffraction at grazing incidence the frequency domain solution is immediately attained along the shadow boundary after the arrival of the leading wavefront. The case of non-grazing incidence is also considered.

The use of scenarios as the basis for combined assessment of climate change mitigation and adaptation

Scenarios are used to explore the consequences of different adaptation and mitigation strategies under uncertainty. In this paper, two scenarios are used to explore developments with (1) no mitigation leading to an increase of global mean temperature of 4 °C by 2100 and (2) an ambitious mitigation strategy leading to 2 °C increase by 2100. For the second scenario, uncertainties in the climate system imply that a global mean temperature increase of 3 °C or more cannot be ruled out. Our analysis shows that, in many cases, adaptation and mitigation are not trade-offs but supplements. For example, the number of people exposed to increased water resource stress due to climate change can be substantially reduced in the mitigation scenario, but adaptation will still be required for the remaining large numbers of people exposed to increased stress. Another example is sea level rise, for which, from a global and purely monetary perspective, adaptation (up to 2100) seems more effective than mitigation. From the perspective of poorer and small island countries, however, stringent mitigation is necessary to keep risks at manageable levels. For agriculture, only a scenario based on a combination of adaptation and mitigation is able to avoid serious climate change impacts.

Contributing understanding of mitigation options for phosphorus and sediment to a review of the efficacy of contemporary agricultural stewardship measures

Experiences from the Mitigation Options for Phosphorus and Sediment (MOPS) projects, which aim to determine the effectiveness of measures to reduce pollutant loading from agricultural land to surface waters, have been used to contribute to the findings of a recent paper (Kay et al., 2009, Agricultural Systems, 99, 67–75), which reviewed the efficacy of contemporary agricultural stewardship measures for ameliorating the water pollution problems of key concern to the UK water industry. MOPS1 is a recently completed 3-year research project on three different soil types in the UK, which focused on mitigation options for winter cereals. MOPS1 demonstrated that tramlines can be the major pathway for sediment and nutrient transfer from arable hillslopes, and that although minimum tillage, crop residue incorporation, contour cultivation, and beetle banks also have potential to be cost-effective mitigation options, tramline management is the one of the most promising treatments for mitigating diffuse pollution losses, as it was able to reduce sediment and nutrient losses by 72–99% in four out of five site years trialled. Using information from the MOPS projects, this paper builds on the findings of Kay et al. to provide an updated picture of the evidence available and the immediate needs for research in this area.

Extension of explicit formulas in Poissonian white noise analysis using harmonic analysis on configuration spaces

Harmonic analysis on configuration spaces is used in order to extend explicit expressions for the images of creation, annihilation, and second quantization operators in L2-spaces with respect to Poisson point processes to a set of functions larger than the space obtained by directly using chaos expansion. This permits, in particular, to derive an explicit expression for the generator of the second quantization of a sub-Markovian contraction semigroup on a set of functions which forms a core of the generator.

Determining the cost of in-field mitigation options to reduce sediment and phosphorus loss

The Mitigation Options for Phosphorus and Sediment (MOPS) project investigated the effectiveness of within-field control measures (tramline management, straw residue management, type of cultivation and direction, and vegetative buffers) in terms of mitigating sediment and phosphorus loss from winter-sown combinable cereal crops using three case study sites. To determine the cost of the approaches, simple financial spreadsheet models were constructed at both farm and regional levels. Taking into account crop areas, crop rotation margins per hectare were calculated to reflect the costs of crop establishment, fertiliser and agro-chemical applications, harvesting, and the associated labour and machinery costs. Variable and operating costs associated with each mitigation option were then incorporated to demonstrate the impact on the relevant crop enterprise and crop rotation margins. These costs were then compared to runoff, sediment and phosphorus loss data obtained from monitoring hillslope-length scale field plots. Each of the mitigation options explored in this study had potential for reducing sediment and phosphorus losses from arable land under cereal crops. Sediment losses were reduced from between 9 kg ha−1 to as much as 4780 kg ha−1 with a corresponding reduction in phosphorus loss from 0.03 kg ha−1 to 2.89 kg ha−1. In percentage terms reductions of phosphorus were between 9% and 99%. Impacts on crop rotation margins also varied. Minimum tillage resulted in cost savings (up to £50 ha−1) whilst other options showed increased costs (up to £19 ha−1 for straw residue incorporation). Overall, the results indicate that each of the options has potential for on-farm implementation. However, tramline management appeared to have the greatest potential for reducing runoff, sediment, and phosphorus losses from arable land (between 69% and 99%) and is likely to be considered cost-effective with only a small additional cost of £2–4 ha−1, although further work is needed to evaluate alternative tramline management methods. Tramline management is also the only option not incorporated within current policy mechanisms associated with reducing soil erosion and phosphorus loss and in light of its potential is an approach that should be encouraged once further evidence is available.

Stability results for the time-harmonic Maxwell equations with impedance boundary conditions

We consider the time-harmonic Maxwell equations with constant coefficients in a bounded, uniformly star-shaped polyhedron. We prove wavenumber-explicit norm bounds for weak solutions. This result is pivotal for convergence proofs in numerical analysis and may be a tool in the analysis of electromagnetic boundary integral operators.

Error analysis of Trefftz-discontinuous Galerkin methods for the time-harmonic Maxwell equations

In this paper, we extend to the time-harmonic Maxwell equations the p-version analysis technique developed in [R. Hiptmair, A. Moiola and I. Perugia, Plane wave discontinuous Galerkin methods for the 2D Helmholtz equation: analysis of the p-version, SIAM J. Numer. Anal., 49 (2011), 264-284] for Trefftz-discontinuous Galerkin approximations of the Helmholtz problem. While error estimates in a mesh-skeleton norm are derived parallel to the Helmholtz case, the derivation of estimates in a mesh-independent norm requires new twists in the duality argument. The particular case where the local Trefftz approximation spaces are built of vector-valued plane wave functions is considered, and convergence rates are derived.

Can REDD+ reconcile local priorities and needs with global mitigation benefits? lessons from Angai Forest, Tanzania

The scope of the reducing emissions from deforestation and forest degradation (REDD) mechanism has broadened REDD+ to accommodate different country interests such as natural forests, protected areas, as well as forests under community-based management. In Tanzania the REDD+ mechanism is still under development and pilot projects are at an early stage. In this paper, we seek to understand how local priorities and needs could be met in REDD+ implementation and how these expectations match with global mitigation benefits. We examine the local priorities and needs in the use of land and forest resources in the Angai Villages Land Forest Reserve (AVLFR) in the Liwale District of Lindi Region in Tanzania. Primary data was collected in two villages, Mihumo and Lilombe, using semistructured key informant interviews and participatory rural appraisal methods. In addition, the key informant interviews were conducted with other village, district, and national level actors, as well as international donors. Findings show that in the two communities REDD+ is seen as something new and is generating new expectations among communities. However, the Angai villagers highlight three key priorities that have yet to be integrated into the design of REDD+: water scarcity, rural development, and food security. At the local level improved forest governance and sustainable management of forest resources have been identified as one way to achieve livelihood diversification. Although the national goals of REDD+ include poverty reduction, these goals are not necessarily conducive to the goals of these communities. There exist both structural and cultural limits to the ability of the Angai villages to implement these goals and to improve forestry governance. Given the vulnerability to current and future climate variability and change it will be important to consider how the AVLFR will be managed and for whose benefit?