An algebraic multigrid method for Q2−Q1Q_2-Q_1 mixed discretizations of the Navier-Stokes equations

Algebraic multigrid (AMG) preconditioners are considered for discretized systems of partial differential equations (PDEs) where unknowns associated with different physical quantities are not necessarily co-located at mesh points. Specifically, we investigate a $Q_2-Q_1$ mixed finite element discretization of the incompressible Navier-Stokes equations where the number of velocity nodes is much greater than the number of pressure nodes. Consequently, some velocity degrees-of-freedom (dofs) are defined at spatial locations where there are no corresponding pressure dofs. Thus, AMG approaches leveraging this co-located structure are not applicable. This paper instead proposes an automatic AMG coarsening that mimics certain pressure/velocity dof relationships of the $Q_2-Q_1$ discretization. The main idea is to first automatically define coarse pressures in a somewhat standard AMG fashion and then to carefully (but automatically) choose coarse velocity unknowns so that the spatial location relationship between pressure and velocity dofs resembles that on the finest grid. To define coefficients within the inter-grid transfers, an energy minimization AMG (EMIN-AMG) is utilized. EMIN-AMG is not tied to specific coarsening schemes and grid transfer sparsity patterns, and so it is applicable to the proposed coarsening. Numerical results highlighting solver performance are given on Stokes and incompressible Navier-Stokes problems.Comment: Submitted to a journa

Assessing architectural evolution: A case study

This is the post-print version of the Article. The official published can be accessed from the link below - Copyright @ 2011 SpringerThis paper proposes to use a historical perspective on generic laws, principles, and guidelines, like Lehman’s software evolution laws and Martin’s design principles, in order to achieve a multi-faceted process and structural assessment of a system’s architectural evolution. We present a simple structural model with associated historical metrics and visualizations that could form part of an architect’s dashboard. We perform such an assessment for the Eclipse SDK, as a case study of a large, complex, and long-lived system for which sustained effective architectural evolution is paramount. The twofold aim of checking generic principles on a well-know system is, on the one hand, to see whether there are certain lessons that could be learned for best practice of architectural evolution, and on the other hand to get more insights about the applicability of such principles. We find that while the Eclipse SDK does follow several of the laws and principles, there are some deviations, and we discuss areas of architectural improvement and limitations of the assessment approach

An analytical framework for the performance evaluation of proximity-aware structured overlays

In this paper, we present an analytical study of proximity-aware structured peer-to-peer networks under churn. We use a master-equation-based approach, which is used traditionally in non-equilibrium statistical mechanics to describe steady-state or transient phenomena. In earlier work we have demonstrated that this methodology is in fact also well suited to describing structured overlay networks under churn, by showing how we can accurately predict the average number of hops taken by a lookup, for any value of churn, for the Chord system. In this paper, we extend the analysis so as to also be able to predict lookup latency, given an average latency for the links in the network. Our results show that there exists a region in the parameter space of the model, depending on churn, the number of nodes, the maintenance rates and the delays in the network, when the network cannot function as a small world graph anymore, due to the farthest connections of a node always being wrong or dead. We also demonstrate how it is possible to analyse proximity neighbour selection or proximity route selection within this formalism