On a general implementation of hh- and pp-adaptive curl-conforming finite elements

Edge (or N\'ed\'elec) finite elements are theoretically sound and widely used by the computational electromagnetics community. However, its implementation, specially for high order methods, is not trivial, since it involves many technicalities that are not properly described in the literature. To fill this gap, we provide a comprehensive description of a general implementation of edge elements of first kind within the scientific software project FEMPAR. We cover into detail how to implement arbitrary order (i.e., $p$-adaptive) elements on hexahedral and tetrahedral meshes. First, we set the three classical ingredients of the finite element definition by Ciarlet, both in the reference and the physical space: cell topologies, polynomial spaces and moments. With these ingredients, shape functions are automatically implemented by defining a judiciously chosen polynomial pre-basis that spans the local finite element space combined with a change of basis to automatically obtain a canonical basis with respect to the moments at hand. Next, we discuss global finite element spaces putting emphasis on the construction of global shape functions through oriented meshes, appropriate geometrical mappings, and equivalence classes of moments, in order to preserve the inter-element continuity of tangential components of the magnetic field. Finally, we extend the proposed methodology to generate global curl-conforming spaces on non-conforming hierarchically refined (i.e., $h$-adaptive) meshes with arbitrary order finite elements. Numerical results include experimental convergence rates to test the proposed implementation

Desarrollo local en el pirineo catalán: impulso económico y uso sostenible del territorio

Local development in the Pyrenee (Catalunya): economicprogress and sustainab le land use. It is important to discriminate activities that respectthe natural and social environment from those focused only on economic results,when studying the local sustainable development in the Catalan Pyrenees; territorial impactsin this area derive from the abandonment of some mountain zones and intensive sitelocations in others. Various cases are described; two are detailed. Methodological premisesare presented that define mountain areas as the periphery of the territory; theoretical frameworksare given for the two selected cases. Conclusions include the need to use models suchas “second-best option” to assess the advantage of each activity.Keywords:

Nouvelles ruralités et développement rural. État de la question et application d’une nouvelle méthode d’analyse en Catalogne: le «rural web»

El present article comença amb un estat de la qüestió sobre el nou paradigma de desenvolupament rural, al qual segueix una crítica a la literatura revisada per elaborar-lo. A continuació, presentem el model de la rural web (Ploeg i Marsden, 2008), un marc teòric innovador sobre el desenvolupament rural, i que alhora pot ser utilitzat com a eina d’anàlisi i de diagnosi. Tot seguit, expliquem com hem aplicat aquest model, a partir de qüestionaris i grups focals, per tal d’analitzar el procés de desenvolupament que ha viscut la comarca del Priorat des dels anys vuitanta fins a l’actualitat. Els resultats obtinguts ens permeten concloure que el model de la rural web constitueix una eina útil per comprendre en profunditat un procés de desenvolupament rural, per bé que no s’ha d’entendre com un model rígid i incontestable.This article starts with the state of the question of the new paradigm of rural development, which is followed by a critical assessment of the literature reviewed for its elaboration. Next, we present the model of the ‘rural web’ (Ploeg and Marsden, 2008), an innovative theoretical framework about rural development which can also be used as an analytical or diagnostic tool. Afterwards, we explain how we applied this model using questionnaires and focus groups to analyse the process of development that has undergone the county of Priorat (SW Catalonia) from the 1980s until now. The results obtained allow us to conclude that the model of the ‘rural web’ is a useful tool for an in-depth understanding of a rural development process, even though it cannot be understood as a rigid or incontestable model.El presente artículo empieza con un estado de la cuestión sobre el nuevo paradigma de desarrollo rural, al que sigue una crítica a la literatura revisada para su elaboración. A continuación, presentamos el modelo de la rural web (Ploeg y Marsden, 2008), un marco teórico innovador sobre el desarrollo rural, y que a la vez puede ser utilizado como herramienta de análisis y de diagnosis. Más adelante, explicamos cómo hemos aplicado este modelo, a partir de cuestionarios y grupos focales, para analizar el proceso de desarrollo que ha vivido la comarca del Priorat (Tarragona) desde los años ochenta hasta la actualidad. Los resultados obtenidos nos permiten concluir que el modelo de la rural web constituye una herramienta útil para comprender en profundidad un proceso de desarrollo rural, aunque no debe ser entendido como un modelo rígido e incontestable.Cet article présente un état de la question du nouveau paradigme du développement rural, suivi d’une évaluation critique de la littérature examinée pour son élaboration. Ensuite, nous présentons le modèle du rural web (Ploeg et Marsden, 2008), un cadre théorique innovant sur le développement rural qui peut également être utilisé comme outil d’analyse ou de diagnostic. À continuation, nous expliquons comment nous avons appliqué ce modèle, en utilisant des questionnaires et des groupes de discussion, pour analyser le processus de développement qui a affecté la région du Priorat (sud-ouest de la Catalogne) à partir des années 1980 jusqu’à nos jours. Les résultats obtenus nous permettent de conclure que le modèle du rural web est un instrument utile pour une compréhension en profondeur d’un processus de développement rural, bien qu’il ne puisse pas être considéré comme un modèle rigide ou incontestable

Balancing domain decomposition by constraints associated with subobjects

A simple variant of the BDDC preconditioner in which constraints are imposed on a selected set of subobjects (subdomain subedges, subfaces and vertices between pairs of subedges) is presented. We are able to show that the condition number of the preconditioner is bounded by C 1 + log(L/h)2, where C is a constant, and h and L are the characteristic sizes of the mesh and the subobjects, respectively. As L can be chosen almost freely, the condition number can theoretically be as small as O(1). We will discuss the pros and cons of the preconditioner and its application to heterogeneous problems. Numerical results on supercomputers are provided

Distributed-memory parallelization of the aggregated unfitted finite element method

The aggregated unfitted finite element method (AgFEM) is a methodology recently introduced in order to address conditioning and stability problems associated with embedded, unfitted, or extended finite element methods. The method is based on removal of basis functions associated with badly cut cells by introducing carefully designed constraints, which results in well-posed systems of linear algebraic equations, while preserving the optimal approximation order of the underlying finite element spaces. The specific goal of this work is to present the implementation and performance of the method on distributed-memory platforms aiming at the efficient solution of large-scale problems. In particular, we show that, by considering AgFEM, the resulting systems of linear algebraic equations can be effectively solved using standard algebraic multigrid preconditioners. This is in contrast with previous works that consider highly customized preconditioners in order to allow one the usage of iterative solvers in combination with unfitted techniques. Another novelty with respect to the methods available in the literature is the problem sizes that can be handled with the proposed approach. While most of previous references discussing linear solvers for unfitted methods are based on serial non-scalable algorithms, we propose a parallel distributed-memory method able to efficiently solve problems at large scales. This is demonstrated by means of a weak scaling test defined on complex 3D domains up to 300M degrees of freedom and one billion cells on 16K CPU cores in the Marenostrum-IV platform. The parallel implementation of the AgFEM method is available in the large-scale finite element package FEMPAR

On the scalability of inexact balancing domain decomposition by constraints with overlapped coarse/fine corrections

In this work, we analyze the scalability of inexact two-level balancing domain decomposition by constraints (BDDC) preconditioners for Krylov subspace iterative solvers, when using a highly scalable asynchronous parallel implementation where fine and coarse correction computations are overlapped in time. This way, the coarse-grid problem can be fully overlapped by fine-grid computations (which are embarrassingly parallel) in a wide range of cases. Further, we consider inexact solvers to reduce the computational cost/complexity and memory consumption of coarse and local problems and boost the scalability of the solver. Out of our numerical experimentation, we conclude that the BDDC preconditioner is quite insensitive to inexact solvers. In particular, one cycle of algebraic multigrid (AMG) is enough to attain algorithmic scalability. Further, the clear reduction of computing time and memory requirements of inexact solvers compared to sparse direct ones makes possible to scale far beyond state-of-the-art BDDC implementations. Excellent weak scalability results have been obtained with the proposed inexact/overlapped implementation of the two-level BDDC preconditioner, up to 93,312 cores and 20 billion unknowns on JUQUEEN. Further, we have also applied the proposed setting to unstructured meshes and partitions for the pressure Poisson solver in the backward-facing step benchmark domain