Parallel implementation of an optimal two level additive Schwarz preconditioner for the 3-D finite element solution of elliptic partial differential equations

This paper presents a description of the extension and parallel implementation of a new two level additive Schwarz (AS) preconditioner for the solution of 3-D elliptic partial differential equations (PDEs). This preconditioner, introduced in Bank et al. (SIAM J. Sci. Comput. 2002; 23: 1818), is based upon the use of a novel form of overlap between the subdomains which makes use of a hierarchy of meshes: with just a single layer of overlapping elements at each level of the hierarchy. The generalization considered here is based upon the restricted AS approach reported in (SIAM J. Sci. Comput. 1999; 21: 792) and the parallel implementation is an extension of work in two dimensions (Concurrency Comput. Practice Experience 2001; 13: 327)

The DUNE-ALUGrid Module

In this paper we present the new DUNE-ALUGrid module. This module contains a major overhaul of the sources from the ALUgrid library and the binding to the DUNE software framework. The main changes include user defined load balancing, parallel grid construction, and an redesign of the 2d grid which can now also be used for parallel computations. In addition many improvements have been introduced into the code to increase the parallel efficiency and to decrease the memory footprint. The original ALUGrid library is widely used within the DUNE community due to its good parallel performance for problems requiring local adaptivity and dynamic load balancing. Therefore, this new model will benefit a number of DUNE users. In addition we have added features to increase the range of problems for which the grid manager can be used, for example, introducing a 3d tetrahedral grid using a parallel newest vertex bisection algorithm for conforming grid refinement. In this paper we will discuss the new features, extensions to the DUNE interface, and explain for various examples how the code is used in parallel environments.Comment: 25 pages, 11 figure

Scalable domain decomposition methods for finite element approximations of transient and electromagnetic problems

The main object of study of this thesis is the development of scalable and robust solvers based on domain decomposition (DD) methods for the linear systems arising from the finite element (FE) discretization of transient and electromagnetic problems. The thesis commences with a theoretical review of the curl-conforming edge (or Nédélec) FEs of the first kind and a comprehensive description of a general implementation strategy for h- and p- adaptive elements of arbitrary order on tetrahedral and hexahedral non-conforming meshes. Then, a novel balancing domain decomposition by constraints (BDDC) preconditioner that is robust for multi-material and/or heterogeneous problems posed in curl-conforming spaces is presented. The new method, in contrast to existent approaches, is based on the definition of the ingredients of the preconditioner according to the physical coefficients of the problem and does not require spectral information. The result is a robust and highly scalable preconditioner that preserves the simplicity of the original BDDC method. When dealing with transient problems, the time direction offers itself an opportunity for further parallelization. Aiming to design scalable space-time solvers, first, parallel-in-time parallel methods for linear and non-linear ordinary differential equations (ODEs) are proposed, based on (non-linear) Schur complement efficient solvers of a multilevel partition of the time interval. Then, these ideas are combined with DD concepts in order to design a two-level preconditioner as an extension to space-time of the BDDC method. The key ingredients for these new methods are defined such that they preserve the time causality, i.e., information only travels from the past to the future. The proposed schemes are weakly scalable in time and space-time, i.e., one can efficiently exploit increasing computational resources to solve more time steps in (approximately) the same time-to-solution. All the developments presented herein are motivated by the driving application of the thesis, the 3D simulation of the low-frequency electromagnetic response of High Temperature Superconductors (HTS). Throughout the document, an exhaustive set of numerical experiments, which includes the simulation of a realistic 3D HTS problem, is performed in order to validate the suitability and assess the parallel performance of the High Performance Computing (HPC) implementation of the proposed algorithms.L’objecte principal d’estudi d’aquesta tesi és el desenvolupament de solucionadors escalables i robustos basats en mètodes de descomposició de dominis (DD) per a sistemes lineals que sorgeixen en la discretització mitjançant elements finits (FE) de problemes transitoris i electromagnètics. La tesi comença amb una revisió teòrica dels FE d’eix (o de Nédélec) de la primera família i una descripció exhaustiva d’una estratègia d’implementació general per a elements h- i p-adaptatius d’ordre arbitrari en malles de tetraedres i hexaedres noconformes. Llavors, es presenta un nou precondicionador de descomposició de dominis balancejats per restricció (BDDC) que és robust per a problemes amb múltiples materials i/o heterogenis definits en espais curl-conformes. El nou mètode, en contrast amb els enfocaments existents, està basat en la definició dels ingredients del precondicionador segons els coeficients físics del problema i no requereix informació espectral. El resultat és un precondicionador robust i escalable que preserva la simplicitat del mètode original BDDC. Quan tractem amb problemes transitoris, la direcció temporal ofereix ella mateixa l’oportunitat de seguir explotant paral·lelisme. Amb l’objectiu de dissenyar precondicionadors en espai-temps, primer, proposem solucionadors paral·lels en temps per equacions diferencials lineals i no-lineals, basats en un solucionador eficient del complement de Schur d’una partició multinivell de l’interval de temps. Seguidament, aquestes idees es combinen amb conceptes de DD amb l’objectiu de dissenyar precondicionadors com a extensió a espai-temps dels mètodes de BDDC. Els ingredients clau d’aquests nous mètodes es defineixen de tal manera que preserven la causalitat del temps, on la informació només viatja de temps passats a temps futurs. Els esquemes proposats són dèbilment escalables en temps i en espai-temps, és a dir, es poden explotar eficientment recursos computacionals creixents per resoldre més passos de temps en (aproximadament) el mateix temps transcorregut de càlcul. Tots els desenvolupaments presentats aquí són motivats pel problema d’aplicació de la tesi, la simulació de la resposta electromagnètica de baixa freqüència dels superconductors d’alta temperatura (HTS) en 3D. Al llarg del document, es realitza un conjunt exhaustiu d’experiments numèrics, els quals inclouen la simulació d’un problema de HTS realista en 3D, per validar la idoneïtat i el rendiment paral·lel de la implementació per a computació d’alt rendiment dels algorismes proposatsPostprint (published version