A three-level BDDC algorithm for mortar discretizations

This is the published version, also available here: http://dx.doi.org/10.1137/07069081X.In this paper, a three-level balancing domain decomposition by constraints (BDDC) algorithm is developed for the solutions of large sparse algebraic linear systems arising from the mortar discretization of elliptic boundary value problems. The mortar discretization is considered on geometrically nonconforming subdomain partitions. In two-level BDDC algorithms, the coarse problem needs to be solved exactly. However, its size will increase with the increase of the number of the subdomains. To overcome this limitation, the three-level algorithm solves the coarse problem inexactly while a good rate of convergence is maintained. This is an extension of previous work: the three-level BDDC algorithms for standard finite element discretization. Estimates of the condition numbers are provided for the three-level BDDC method, and numerical experiments are also discussed

Isogeometric BDDC Preconditioning with Deluxe Scaling

A balancing domain decomposition by constraints (BDDC) preconditioner with a novel scaling, introduced by Dohrmann for problems with more than one variable coefficient and here denoted as deluxe scaling, is extended to isogeometric analysis of scalar elliptic problems. This new scaling turns out to be more powerful than the standard ?- and stiffness scalings considered in a previous isogeometric BDDC study. Our h-analysis shows that the condition number of the resulting deluxe BDDC preconditioner is scalable with a quasi-optimal polylogarithmic bound which is also independent of coefficient discontinuities across subdomain interfaces. Extensive numerical experiments support the theory and show that the deluxe scaling yields a remarkable improvement over the older scalings, in particular for large isogeometric polynomial degree and high regularity

Robust exact and inexact FETI-DP methods with applications to elasticity

Gebietszerlegungsverfahren sind parallele, iterative Lösungsverfahren für grosse Gleichungssysteme, die bei der Diskretisierung von partiellen Differentialgleichungen, etwa aus der Strukturmechanik, entstehen. In dieser Arbeit werden duale, iterative Substrukturierungsverfahren vom FETI-DP-Typ (Finite Element Tearing and Interconnecting Dual-Primal) entwickelt und auf elliptische partielle Differentialgleichungen zweiter Ordnung angewandt. Insbesondere wird versucht, robuste Verfahren für homogene und heterogene Elastizitaetsprobleme zu entwickeln. Ebenso werden neue, inexakte FETI-DP-Verfahren vorgestellt, die eine inexakte Lösung des Grobgitterproblems und/oder der Teilgebietsprobleme erlauben. Es wird gezeigt, dass die neuen Algorithmen unter bestimmten Voraussetzungen Abschätzungen der gleichen asymptotischen Güte wie das klassische, exakte FETI-DP-Verfahren erfüllen. Parallele Resultate unter Verwendung von algebraischen Mehrgitter für das Grobgitterproblem zeigen die verbesserte Skalierbarkeit der neuen Algorithmen.Domain decomposition methods are fast parallel solvers for large equation systems arising from the discretisation of partial differential equations, e.g. from structural mechanics. In this work, dual iterative substructuring methods of the FETI-DP (Finite Element Tearing and Interconnecting Dual-Primal) type are developed and applied to second order elliptic problems with emphasis on elasticity. An attempt is made to develop robust methods for homogeneous and heterogeneous problems. New inexact FETI-DP methods are also introduced that allow for inexact coarse problem solvers and/or inexact subdomain solvers. It is shown that under certain conditions the new algorithms fulfill the same asymptotic condition number estimate as the traditional, exact FETI-DP methods. Parallel results using algebraic multigrid for the FETI-DP coarse problem show the improved scalability of the new algorithms

Local Fourier analysis for saddle-point problems

The numerical solution of saddle-point problems has attracted considerable interest in recent years, due to their indefiniteness and often poor spectral properties that make efficient solution difficult. While much research already exists, developing efficient algorithms remains challenging. Researchers have applied finite-difference, finite element, and finite-volume approaches successfully to discretize saddle-point problems, and block preconditioners and monolithic multigrid methods have been proposed for the resulting systems. However, there is still much to understand. Magnetohydrodynamics (MHD) models the flow of a charged fluid, or plasma, in the presence of electromagnetic fields. Often, the discretization and linearization of MHD leads to a saddle-point system. We present vector-potential formulations of MHD and a theoretical analysis of the existence and uniqueness of solutions of both the continuum two-dimensional resistive MHD model and its discretization. Local Fourier analysis (LFA) is a commonly used tool for the analysis of multigrid and other multilevel algorithms. We first adapt LFA to analyse the properties of multigrid methods for both finite-difference and finite-element discretizations of the Stokes equations, leading to saddle-point systems. Monolithic multigrid methods, based on distributive, Braess-Sarazin, and Uzawa relaxation are discussed. From this LFA, optimal parameters are proposed for these multigrid solvers. Numerical experiments are presented to validate our theoretical results. A modified two-level LFA is proposed for high-order finite-element methods for the Lapalce problem, curing the failure of classical LFA smoothing analysis in this setting and providing a reliable way to estimate actual multigrid performance. Finally, we extend LFA to analyze the balancing domain decomposition by constraints (BDDC) algorithm, using a new choice of basis for the space of Fourier harmonics that greatly simplifies the application of LFA. Improved performance is obtained for some two- and three-level variants

Domain Decomposition Methods for Elastic Materials with Compressible and Almost Incompressible Components

Domain decomposition methods are iterative methods to solve large systems of equations, obtained, e.g., from finite element discretization. Here, the domain is decomposed into smaller subproblems, which can be solved in parallel. In the first part of this work, new condition number bounds are proven for a FETI-DP type (Finite Element Tearing and Interconnecting Dual-Primal) domain decomposition method for compressible linear elasticity in 3D. Each subdomain may contain an inclusion having different material properties. The condition number bound only depends on the subdomain diameter, the finite element diameter, and the thickness of the compressible hull. It is independent of the material parameters in the inclusions, thus almost incompressible inclusions are also possible. In the second part of this thesis a new coarse space for FETI-DP methods for almost incompressible linear elasticity on the whole domain is presented. This coarse space is much smaller than the standard coarse space for FETI-DP or BDDC methods for almost incompressible linear elasticity.Gebietszerlegungsalgorithmen sind iterative Verfahren zum Lösen großer Gleichungssysteme, die z. B. durch den Finite-Elemente-Ansatz entstehen. Dabei wird das Ausgangsproblem in kleinere Teilprobleme zerlegt, die dann parallel gelöst werden können. Im ersten Teil der Arbeit werden neue Konditionszahlabschätzungen für Gebietszerlegungsverfahren vom FETI-DP- Typ (Finite Element Tearing and Interconnecting Dual-Primal) für kompressible lineare Elastizitätsprobleme in 3D bewiesen, wobei in jedem Gebiet Einschlüsse mit anderen Materialparametern eingebettet sein können. Die Abschätzungen hängen dabei nur von dem typischen Teilgebietsdurchmesser, dem Finite-Elemente-Durchmesser und der Breite einer kompressiblen Hülle ab. Sie ist unabhängig von den Materialparametern in den Einschlüssen. Auch fast-inkompressible Einschlüsse sind möglich. Im zweiten Teil der Arbeit wird ein neuer Grobgitterraum für FETI-DP-Verfahren für fast-inkompressible Elastizität vorgestellt. Dieser Grobgitterraum ist erheblich kleiner als bisher bekannte Grobgitterräume für FETI-DP oder BDDC-Verfahren für fast-inkompressible lineare Elastizität