Parallel unstructured solvers for linear partial differential equations

This thesis presents the development of a parallel algorithm to solve symmetric systems of linear equations and the computational implementation of a parallel partial differential equations solver for unstructured meshes. The proposed method, called distributive conjugate gradient - DCG, is based on a single-level domain decomposition method and the conjugate gradient method to obtain a highly scalable parallel algorithm. An overview on methods for the discretization of domains and partial differential equations is given. The partition and refinement of meshes is discussed and the formulation of the weighted residual method for two- and three-dimensions presented. Some of the methods to solve systems of linear equations are introduced, highlighting the conjugate gradient method and domain decomposition methods. A parallel unstructured PDE solver is proposed and its actual implementation presented. Emphasis is given to the data partition adopted and the scheme used for communication among adjacent subdomains is explained. A series of experiments in processor scalability is also reported. The derivation and parallelization of DCG are presented and the method validated throughout numerical experiments. The method capabilities and limitations were investigated by the solution of the Poisson equation with various source terms. The experimental results obtained using the parallel solver developed as part of this work show that the algorithm presented is accurate and highly scalable, achieving roughly linear parallel speed-up in many of the cases tested

Structural Analysis and Matrix Interpetive System /SAMIS/ program Technical report, Feb. - Aug. 1966

Development of characteristic equations and error analysis for computer programs contained in structural analysis and matrix interpretive syste

Parallel unstructured solvers for linear partial differential equations

This thesis presents the development of a parallel algorithm to solve symmetric systems of linear equations and the computational implementation of a parallel partial differential equations solver for unstructured meshes. The proposed method, called distributive conjugate gradient - DCG, is based on a single-level domain decomposition method and the conjugate gradient method to obtain a highly scalable parallel algorithm. An overview on methods for the discretization of domains and partial differential equations is given. The partition and refinement of meshes is discussed and the formulation of the weighted residual method for two- and three-dimensions presented. Some of the methods to solve systems of linear equations are introduced, highlighting the conjugate gradient method and domain decomposition methods. A parallel unstructured PDE solver is proposed and its actual implementation presented. Emphasis is given to the data partition adopted and the scheme used for communication among adjacent subdomains is explained. A series of experiments in processor scalability is also reported. The derivation and parallelization of DCG are presented and the method validated throughout numerical experiments. The method capabilities and limitations were investigated by the solution of the Poisson equation with various source terms. The experimental results obtained using the parallel solver developed as part of this work show that the algorithm presented is accurate and highly scalable, achieving roughly linear parallel speed-up in many of the cases tested.EThOS - Electronic Theses Online ServiceGBUnited Kingdo

Seventh Copper Mountain Conference on Multigrid Methods

The Seventh Copper Mountain Conference on Multigrid Methods was held on 2-7 Apr. 1995 at Copper Mountain, Colorado. This book is a collection of many of the papers presented at the conference and so represents the conference proceedings. NASA Langley graciously provided printing of this document so that all of the papers could be presented in a single forum. Each paper was reviewed by a member of the conference organizing committee under the coordination of the editors. The multigrid discipline continues to expand and mature, as is evident from these proceedings. The vibrancy in this field is amply expressed in these important papers, and the collection shows its rapid trend to further diversity and depth

Das unstetige Galerkinverfahren für Strömungen mit freier Oberfläche und im Grundwasserbereich in geophysikalischen Anwendungen

Free surface flows and subsurface flows appear in a broad range of geophysical applications and in many environmental settings situations arise which even require the coupling of free surface and subsurface flows. Many of these application scenarios are characterized by large domain sizes and long simulation times. Hence, they need considerable amounts of computational work to achieve accurate solutions and the use of efficient algorithms and high performance computing resources to obtain results within a reasonable time frame is mandatory. Discontinuous Galerkin methods are a class of numerical methods for solving differential equations that share characteristics with methods from the finite volume and finite element frameworks. They feature high approximation orders, offer a large degree of flexibility, and are well-suited for parallel computing. This thesis consists of eight articles and an extended summary that describe the application of discontinuous Galerkin methods to mathematical models including free surface and subsurface flow scenarios with a strong focus on computational aspects. It covers discretization and implementation aspects, the parallelization of the method, and discrete stability analysis of the coupled model.Für viele geophysikalische Anwendungen spielen Strömungen mit freier Oberfläche und im Grundwasserbereich oder sogar die Kopplung dieser beiden eine zentrale Rolle. Oftmals charakteristisch für diese Anwendungsszenarien sind große Rechengebiete und lange Simulationszeiten. Folglich ist das Berechnen akkurater Lösungen mit beträchtlichem Rechenaufwand verbunden und der Einsatz effizienter Lösungsverfahren sowie von Techniken des Hochleistungsrechnens obligatorisch, um Ergebnisse innerhalb eines annehmbaren Zeitrahmens zu erhalten. Unstetige Galerkinverfahren stellen eine Gruppe numerischer Verfahren zum Lösen von Differentialgleichungen dar, und kombinieren Eigenschaften von Methoden der Finiten Volumen- und Finiten Elementeverfahren. Sie ermöglichen hohe Approximationsordnungen, bieten einen hohen Grad an Flexibilität und sind für paralleles Rechnen gut geeignet. Diese Dissertation besteht aus acht Artikeln und einer erweiterten Zusammenfassung, in diesen die Anwendung unstetiger Galerkinverfahren auf mathematische Modelle inklusive solcher für Strömungen mit freier Oberfläche und im Grundwasserbereich beschrieben wird. Die behandelten Themen umfassen Diskretisierungs- und Implementierungsaspekte, die Parallelisierung der Methode sowie eine diskrete Stabilitätsanalyse des gekoppelten Modells