Analysis of Schwarz methods for a hybridizable discontinuous Galerkin discretization

Schwarz methods are attractive parallel solvers for large scale linear systems obtained when partial differential equations are discretized. For hybridizable discontinuous Galerkin (HDG) methods, this is a relatively new field of research, because HDG methods impose continuity across elements using a Robin condition, while classical Schwarz solvers use Dirichlet transmission conditions. Robin conditions are used in optimized Schwarz methods to get faster convergence compared to classical Schwarz methods, and this even without overlap, when the Robin parameter is well chosen. We present in this paper a rigorous convergence analysis of Schwarz methods for the concrete case of hybridizable interior penalty (IPH) method. We show that the penalization parameter needed for convergence of IPH leads to slow convergence of the classical additive Schwarz method, and propose a modified solver which leads to much faster convergence. Our analysis is entirely at the discrete level, and thus holds for arbitrary interfaces between two subdomains. We then generalize the method to the case of many subdomains, including cross points, and obtain a new class of preconditioners for Krylov subspace methods which exhibit better convergence properties than the classical additive Schwarz preconditioner. We illustrate our results with numerical experiments.Comment: 25 pages, 5 figures, 3 tables, accepted for publication in SINU

Cracked but not broken: the continued gender gap in senior administrative positions

Equality for women (and men) was enshrined in the Treaty of Rome in 1957. Legislatively this has meant that women have been treated in the same way as men, and post-feminist theory suggests that equality battles have been won and that women and men now have equal opportunities. However, when looking into the attainment of women in HE, the concept of equal opportunities seems questionable. Female administrative staff have outnumbered male administrative staff in universities for at least sixteen years. However, there is still an unbalanced gender profile at senior, ie above G9 (spine point >51 on the new single pay framework [UCU 2004]). If for professional managers at levels below that of Grade 10 women outweigh men by on average 62% why at levels at G10 and above do women represent on average only 30% of staff

Hybrid organizations as a strategy for supporting new product development

Alliances between large, well-established corporations and highly creative small companies or consultancies can be an effective method for promoting innovation

Cross-Points in Domain Decomposition Methods with a Finite Element Discretization

Non-overlapping domain decomposition methods necessarily have to exchange Dirichlet and Neumann traces at interfaces in order to be able to converge to the underlying mono-domain solution. Well known such non-overlapping methods are the Dirichlet-Neumann method, the FETI and Neumann-Neumann methods, and optimized Schwarz methods. For all these methods, cross-points in the domain decomposition configuration where more than two subdomains meet do not pose any problem at the continuous level, but care must be taken when the methods are discretized. We show in this paper two possible approaches for the consistent discretization of Neumann conditions at cross-points in a Finite Element setting

Designing Scalable Business Models

Digital business models are often designed for rapid growth, and some relatively young companies have indeed achieved global scale. However despite the visibility and importance of this phenomenon, analysis of scale and scalability remains underdeveloped in management literature. When it is addressed, analysis of this phenomenon is often over-influenced by arguments about economies of scale in production and distribution. To redress this omission, this paper draws on economic, organization and technology management literature to provide a detailed examination of the sources of scaling in digital businesses. We propose three mechanisms by which digital business models attempt to gain scale: engaging both non- paying users and paying customers; organizing customer engagement to allow self- customization; and orchestrating networked value chains, such as platforms or multi-sided business models. Scaling conditions are discussed, and propositions developed and illustrated with examples of big data entrepreneurial firms

A new Algorithm Based on Factorization for Heterogeneous Domain Decomposition

Often computational models are too expensive to be solved in the entire domain of simulation, and a cheaper model would suffice away from the main zone of interest. We present for the concrete example of an evolution problem of advection reaction diffusion type a heterogeneous domain decomposition algorithm which allows us to recover a solution that is very close to the solution of the fully viscous problem, but solves only an inviscid problem in parts of the domain. Our new algorithm is based on the factorization of the underlying differential operator, and we therefore call it factorization algorithm. We give a detailed error analysis, and show that we can obtain approximations in the viscous region which are much closer to the viscous solution in the entire domain of simulation than approximations obtained by other heterogeneous domain decomposition algorithms from the literature.Comment: 23 page

Dirichlet-Neumann and Neumann-Neumann Waveform Relaxation for the Wave Equation

We present a Waveform Relaxation (WR) version of the Dirichlet-Neumann and Neumann-Neumann algorithms for the wave equation in space time. Each method is based on a non-overlapping spatial domain decomposition, and the iteration involves subdomain solves in space time with corresponding interface condition, followed by a correction step. Using a Laplace transform argument, for a particular relaxation parameter, we prove convergence of both algorithms in a finite number of steps for finite time intervals. The number of steps depends on the size of the subdomains and the time window length on which the algorithms are employed. We illustrate the performance of the algorithms with numerical results, and also show a comparison with classical and optimized Schwarz WR methods.Comment: 8 pages, 6 figures, presented in 22nd International conference on Domain Decomposition Methods, to appear in Domain Decomposition in Science and Engineering XXII, LNCSE, Springer-Verlag 201