134 research outputs found
FETI-DP algorithms for 2D Biot model with discontinuous Galerkin discretization
Dual-Primal Finite Element Tearing and Interconnecting (FETI-DP) algorithms
are developed for a 2D Biot model. The model is formulated with mixed-finite
elements as a saddle-point problem. The displacement and the Darcy
flux flow are represented with piecewise continuous elements
and pore-pressure with piecewise constant elements, {\it i.e.},
overall three fields with a stabilizing term. We have tested the functionality
of FETI-DP with Dirichlet preconditioners. Numerical experiments show a
signature of scalability of the resulting parallel algorithm in the
compressible elasticity with permeable Darcy flow as well as almost
incompressible elasticity.Comment: Accepted to the 27th International Conference on Domain Decomposition
Methods (DD27), 8 pages. arXiv admin note: text overlap with arXiv:2211.1502
A Nonoverlapping Domain Decomposition Method for Incompressible Stokes Equations with Continuous Pressures
This is the publisher's version, also available electronically from http://epubs.siam.org/doi/abs/10.1137/120861503A nonoverlapping domain decomposition algorithm is proposed to solve the linear system arising from mixed finite element approximation of incompressible Stokes equations. A continuous finite element space for the pressure is used. In the proposed algorithm, Lagrange multipliers are used to enforce continuity of the velocity component across the subdomain boundary. The continuity of the pressure component is enforced in the primal form, i.e., neighboring subdomains share the same pressure degrees of freedom on the subdomain interface and no Lagrange multipliers are needed. After eliminating all velocity variables and the independent subdomain interior parts of the pressures, a symmetric positive semidefinite linear system for the subdomain boundary pressures and the Lagrange multipliers is formed and solved by a preconditioned conjugate gradient method. A lumped preconditioner is studied and the condition number bound of the preconditioned operator is proved to be independent of the number of subdomains for fixed subdomain problem size. Numerical experiments demonstrate the convergence rate of the proposed algorithm
A FETI-DP TYPE DOMAIN DECOMPOSITION ALGORITHM FOR THREE-DIMENSIONAL INCOMPRESSIBLE STOKES EQUATIONS
The FETI-DP (dual-primal finite element tearing and interconnecting) algorithms,
proposed by the authors in [SIAM J. Numer. Anal., 51 (2013), pp. 1235–1253] and [Internat. J.
Numer. Methods Engrg., 94 (2013), pp. 128–149] for solving incompressible Stokes equations, are
extended to three-dimensional problems. A new analysis of the condition number bound for using
the Dirichlet preconditioner is given. The algorithm and analysis are valid for mixed finite
elements with both continuous and discontinuous pressures. An advantage of this new analysis is
that the numerous coarse level velocity components, required in the previous analysis to enforce the
divergence-free subdomain boundary velocity conditions, are no longer needed. This greatly reduces
the size of the coarse level problem in the algorithm, especially for three-dimensional problems. The
coarse level velocity space can be chosen as simple as those coarse spaces for solving scalar elliptic
problems corresponding to each velocity component. Both the Dirichlet and lumped preconditioners
are analyzed using the same framework in this new analysis. Their condition number bounds are
proved to be independent of the number of subdomains for fixed subdomain problem size. Numerical
experiments in both two and three dimensions, using mixed finite elements with both continuous
and discontinuous pressures, demonstrate the convergence rate of the algorithms
Stabilization of a non standard FETI-DP mortar method for the Stokes problem
In a recent paper [E. Chacón Vera and D. Franco Coronil, J. Numer. Math. 20 (2012) 161–182.] a non standard mortar method for incompressible Stokes problem was introduced where the use of the trace spaces H1/2 and H1/2 00 and a direct computation of the pairing of the trace spaces with their duals are the main ingredients. The importance of the reduction of the number of degrees of freedom leads naturally to consider the stabilized version and this is the results we present in this work. We prove that the standard Brezzi–Pitkaranta stabilization technique is available and that it works well with this mortar method. Finally, we present some numerical tests to illustrate this behaviour.Ministerio de Ciencia e InnovaciónJunta de Andaluci
Software concepts and algorithms for an efficient and scalable parallel finite element method
Software packages for the numerical solution of partial differential equations (PDEs) using the finite element method are important in different fields of research. The basic data structures and algorithms change in time, as the user\'s requirements are growing and the software must efficiently use the newest highly parallel computing systems. This is the central point of this work.
To make efficiently use of parallel computing systems with growing number of independent basic computing units, i.e.~CPUs, we have to combine data structures and algorithms from different areas of mathematics and computer science. Two crucial parts are a distributed mesh and parallel solver for linear systems of equations. For both there exists multiple independent approaches. In this work we argue that it is necessary to combine both of them to allow for an efficient and scalable implementation of the finite element method. First, we present concepts, data structures and algorithms for distributed meshes, which allow for local refinement. The central point of our presentation is to provide arbitrary geometrical information of the mesh and its distribution to the linear solver.
A large part of the overall computing time of the finite element method is spend by the linear solver. Thus, its parallelization is of major importance. Based on the presented concept for distributed meshes, we preset several different linear solver methods. Hereby we concentrate on general purpose linear solver, which makes only little assumptions about the systems to be solver. For this, a new FETI-DP (Finite Element Tearing and Interconnect - Dual Primal) method is proposed. Those the standard FETI-DP method is quasi optimal from a mathematical point of view, its not possible to implement it efficiently for a large number of processors (> 10,000). The main reason is a relatively small but globally distributed coarse mesh problem. To circumvent this problem, we propose a new multilevel FETI-DP method which hierarchically decompose the coarse grid problem. This leads to a more local communication pattern for solver the coarse grid problem and makes it possible to scale for a large number of processors.
Besides the parallelization of the finite element method, we discuss an approach to speed up serial computations of existing finite element packages. In many computations the PDE to be solved consists of more than one variable. This is especially the case in multi-physics modeling. Observation show that in many of these computation the solution structure of the variables is different. But in the standard finite element method, only one mesh is used for the discretization of all variables. We present a multi-mesh finite element method, which allows to discretize a system of PDEs with two independently refined meshes.Softwarepakete zur numerischen Lösung partieller Differentialgleichungen mit Hilfe der Finiten-Element-Methode sind in vielen Forschungsbereichen ein wichtiges Werkzeug. Die dahinter stehenden Datenstrukturen und Algorithmen unterliegen einer ständigen Neuentwicklung um den immer weiter steigenden Anforderungen der Nutzergemeinde gerecht zu werden und um neue, hochgradig parallel Rechnerarchitekturen effizient nutzen zu können. Dies ist auch der Kernpunkt dieser Arbeit.
Um parallel Rechnerarchitekturen mit einer immer höher werdenden Anzahl an von einander unabhängigen Recheneinheiten, z.B.~Prozessoren, effizient Nutzen zu können, müssen Datenstrukturen und Algorithmen aus verschiedenen Teilgebieten der Mathematik und Informatik entwickelt und miteinander kombiniert werden. Im Kern sind dies zwei Bereiche: verteilte Gitter und parallele Löser für lineare Gleichungssysteme. Für jedes der beiden Teilgebiete existieren unabhängig voneinander zahlreiche Ansätze. In dieser Arbeit wird argumentiert, dass für hochskalierbare Anwendungen der Finiten-Elemente-Methode nur eine Kombination beider Teilgebiete und die Verknüpfung der darunter liegenden Datenstrukturen eine effiziente und skalierbare Implementierung ermöglicht. Zuerst stellen wir Konzepte vor, die parallele verteile Gitter mit entsprechenden Adaptionstrategien ermöglichen. Zentraler Punkt ist hier die Informationsaufbereitung für beliebige Löser linearer Gleichungssysteme.
Beim Lösen partieller Differentialgleichung mit der Finiten Elemente Methode wird ein großer Teil der Rechenzeit für das Lösen der dabei anfallenden linearen Gleichungssysteme aufgebracht. Daher ist deren Parallelisierung von zentraler Bedeutung. Basierend auf dem vorgestelltem Konzept für verteilten Gitter, welches beliebige geometrische Informationen für die linearen Löser aufbereiten kann, präsentieren wir mehrere unterschiedliche Lösermethoden. Besonders Gewicht wird dabei auf allgemeine Löser gelegt, die möglichst wenig Annahmen über das zu lösende System machen. Hierfür wird die FETI-DP (Finite Element Tearing and Interconnect - Dual Primal) Methode weiterentwickelt. Obwohl die FETI-DP Methode vom mathematischen Standpunkt her als quasi-optimal bezüglich der parallelen Skalierbarkeit gilt, kann sie für große Anzahl an Prozessoren (> 10.000) nicht mehr effizient implementiert werden. Dies liegt hauptsächlich an einem verhältnismäßig kleinem aber global verteilten Grobgitterproblem. Wir stellen eine Multilevel FETI-DP Methode vor, die dieses Problem durch eine hierarchische Komposition des Grobgitterproblems löst. Dadurch wird die Kommunikation entlang des Grobgitterproblems lokalisiert und die Skalierbarkeit der FETI-DP Methode auch für große Anzahl an Prozessoren sichergestellt.
Neben der Parallelisierung der Finiten-Elemente-Methode beschäftigen wir uns in dieser Arbeit mit der Ausnutzung von bestimmten Voraussetzung um auch die sequentielle Effizienz bestehender Implementierung der Finiten-Elemente-Methode zu steigern. In vielen Fällen müssen partielle Differentialgleichungen mit mehreren Variablen gelöst werden. Sehr häufig ist dabei zu beobachten, insbesondere bei der Modellierung mehrere miteinander gekoppelter physikalischer Phänomene, dass die Lösungsstruktur der unterschiedlichen Variablen entweder schwach oder vollständig voneinander entkoppelt ist. In den meisten Implementierungen wird dabei nur ein Gitter zur Diskretisierung aller Variablen des Systems genutzt. Wir stellen eine Finite-Elemente-Methode vor, bei der zwei unabhängig voneinander verfeinerte Gitter genutzt werden können um ein System partieller Differentialgleichungen zu lösen
Stable discretizations and IETI-DP solvers for the Stokes system in multi-patch Isogeometric Analysis
We are interested in a fast solver for the Stokes equations, discretized with
multi-patch Isogeometric Analysis. In the last years, several inf-sup stable
discretizations for the Stokes problem have been proposed, often the analysis
was restricted to single-patch domains. We focus on one of the simplest
approaches, the isogeometric Taylor--Hood element. We show how stability
results for single-patch domains can be carried over to multi-patch domains.
While this is possible, the stability strongly depends on the shape of the
geometry. We construct a Dual-Primal Isogeometric Tearing and Interconnecting
(IETI-DP) solver that does not suffer from that effect. We give a convergence
analysis and provide numerical tests
Multigrid and saddle-point preconditioners for unfitted finite element modelling of inclusions
In this work, we consider the modeling of inclusions in the material using an
unfitted finite element method. In the unfitted methods, structured background
meshes are used and only the underlying finite element space is modified to
incorporate the discontinuities, such as inclusions. Hence, the unfitted
methods provide a more flexible framework for modeling the materials with
multiple inclusions. We employ the method of Lagrange multipliers for enforcing
the interface conditions between the inclusions and matrix, this gives rise to
the linear system of equations of saddle point type. We utilize the Uzawa
method for solving the saddle point system and propose preconditioning
strategies for primal and dual systems.
For the dual systems, we review and compare the preconditioning strategies
that are developed for FETI and SIMPLE methods. While for the primal system, we
employ a tailored multigrid method specifically developed for the unfitted
meshes. Lastly, the comparison between the proposed preconditioners is made
through several numerical experiments
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