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

Europeanisation should meet international constructivism: the Nordic Plus group and the internalisation of political conditionality by France and the United Kingdom

This article is a plausibility probe for the significance of international constructivist ‘mediating factors’ to explain variation in Europeanisation outcomes. It applies a most similar systems design (or Mill's method of difference) to show that the UK has internalised political conditionality to a larger extent than France at least partially because it has been the object of stronger socialisation pressures within the ‘Nordic Plus’ group. The article contributes to the literature on Europeanisation and development cooperation in two important ways. First, it enlarges its scope of analysis, both geographically (beyond new European Union member states) and thematically (beyond simple measures of aid quality and/or quantity). Second, it emphasises the importance of international (versus domestic) mediating factors. The empirical analysis focusses on three cases of aid sanctions in response to human rights abuses and democratic setbacks: Zimbabwe 2002, Madagascar 2009 and Mozambique 2009

Haemodynamic changes associated with atropine/neostigmine administration: A non-invasive investigation

SCOPUS: ar.jFLWNAinfo:eu-repo/semantics/publishe