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

Perfluorinated substances in human food and other sources of human exposure

Perfluorinated compounds (PFCs) are ubiquitous environmental contaminants, which persist and may bioaccumulate through the food chain (Haukås et al. 2007; Martin et al. 2004b; Taniyasu et al. 2003). As a consequence, several PFCs have been detected in different biota worldwide. In recent years, an increasing number of papers report high levels of PFCs in blood, tissues, and breast milk from both occupationally and non-occupationally exposed human populations (Kannan et al. 2004; Kärrman et al. 2007; Olsen et al. 2007). The most important exposure pathways of perfluorinated compounds for humans are thought to be intake of drinking water and food and inhalation of dust (Björklund et al. 2009; Ericson et al. 2008a). Due to the widespread distribution, environmental degradation, and metabolism of the PFCs released into the environment, a very complex exposure situation exists (Fromme et al. 2007a). As a result, the relative contribution to human exposure from different routes or from a single source (e.g., diet) is not yet known. More specifically, it is currently unknown as to what extent exposure to drinking water, food, or dust contributes to the PFCs measured in human breast milk and blood. Moreover, data on levels of PFCs in the human diet are rather scarce (Kärrman et al. 2009; Tittlemier et al. 2006, 2007). Only PFC levels in fish appear to be well documented (Houde et al. 2006). Few studies, however, report the levels of PFCs in drinking water or human food such as vegetables, meat, and eggs (FSA 2006; US EPA 2001). Food processing such as cooking (boiling, baking, or grilling) could alter the concentration of PFCs in food and as a consequence affect the risk for humans

Brominated flame retardants and perfluorinated compounds in indoor dust from homes and offices in Flanders, Belgium

The increasing time spent indoors combined with the abundant usage of diverse indoor chemicals led to concerns involving the impact of these compounds on human health. The current study focused on two groups of important indoor contaminants i.e. Brominated flame retardants (BFRs) and Perfluorinated compounds (PFCs). Concentrations of both compound classes have been measured in Flemish indoor dust samples from homes and offices. Sigma Polybrominated diphenyl ethers (PBDEs) (BDE 47, 99, 100, 154, 153, 197, 196 and 203) and BDE 209 in homes ranged between 4-1214 ng g(-1) dw (median 35) and <5-5295 ng g(-1) dw (median 313). respectively. Hexabromocyclododecane (Sigma HBCD) levels ranged from 5 to 42 692 ng g(-1) dw (median 130), with alpha-HBCD being the major isomer (mean 59%). In addition, tetrabromobisphenol A (TBBPA) ranged between <3 and 419 ng g(-1) dw (median 12). For all BFRs, median levels in office dust were up to an order of magnitude higher than in home dust. Sigma PFCs (sum of perfluorobutane sulfonate (PFBS), perfluorohexane sulfonate (PFHxS), perfluorooctane sulfonic acid (PFOS), perfluorobutanoic acid (PFBA), perfluorohexanoic acid (PFHxA), perfluorooctanoic acid (PFOA), perfluorononanoic acid (PFNA) and perfluorodecanoic acid (PFDA)) concentrations in homes ranged from 0.2 to 336 ng g(-1) (median 3.0 ng g(-1)). Levels in office dust were higher (p < 0.01) than in house dust with Sigma PFCs ranging between 2.2 and 647 ng g(-1) (median 10 ng g(-1)) and median (PFOA) and perfluorooctane sulfonate values of 2.9 and 2.2 ng g(-1), respectively. The congener pattern was dominated by PFOA, followed by PFOS. Calculated human exposure was below the reference dose values set by the US-EPA for BDE 209, HBCD and below the provisional tolerable daily intakes proposed by European Food Safety Authority for PFOS and PFOA