Search

In the past few years, Domain Decomposition Methods (DDM) have emerged as advanced solvers in several areas of computational mechanics. In particular, during the last decade, in the area of solid and structural mechanics, they reached a considerable level of advancement and have been shown to be more efficient than popular solvers, like advanced sparse direct solvers. The present contribution follows the lines of a series of recent publications by the authors on DDM. In these papers, the authors developed a unified theory of primal and dual methods and presented a family of DDM that were shown to be more efficient than previous methods. The present paper extends this work, presenting a new family of related DDM, thus enriching the theory of the relations between primal and dual methods. [ABSTRACT FROM AUTHOR]

Recent progress has been made to more robustly handle the increased complexity of high-order schemes by focusing on the local nature of the discretization. This locality is particularly true for many Discontinuous Galerkin formulations and is the focus of this paper. The contributions of this paper are twofold. First, novel observations regarding various flux representations in the discontinuous Galerkin formulation are highlighted in the context of overlapping Schwarz methods. Second, we conduct additional experiments using high-order elements for the indefinite Helmholtz equation to expose the impact of overlap. [ABSTRACT FROM AUTHOR]

The parareal algorithm is a method to solve time dependent problems parallel in time: it approximates parts of the solution later in time simultaneously to parts of the solution earlier in time. In this paper the relation of the parareal algorithm to space-time multigrid and multiple shooting methods is first briefly discussed. The focus of the paper is on some new convergence results that show superlinear convergence of the algorithm when used on bounded time intervals, and linear convergence for unbounded intervals. [ABSTRACT FROM AUTHOR]

The paper analyzes the approximation of the weak continuity constraint in the mortar method, and provides error estimates in the L2 -norm and H1 broken norm for generic discretization spaces, treating the presence of cross-points in the geometrical decomposition. [ABSTRACT FROM AUTHOR]

This paper addresses scalable, parallel hp-finite/infinite element-based solution of time-harmonic acoustics problems in three-dimensions. We discuss the application of FETI-DP, an iterative domain-decomposition scheme, to both interior and exterior acoustics problems. [ABSTRACT FROM AUTHOR]

In recent years, several extensions of the classical AMG method (see [2] and [10]) to handle more general finite element matrices have been proposed (see, e.g., [3], [9], and [7]). Other extensions are related to the so-called smoothed aggregation method; see e.g., [11] and the papers cited therein. For the most recent versions of both the AMG and smoothed aggregation approaches, we refer to [5] and [4]. [ABSTRACT FROM AUTHOR]

In this paper we design and apply new embedded pairs of Fractional Step Runge-Kutta methods to the efficient solution of multidimensional parabolic problems. These time integrators are combined with a suitable splitting of the elliptic operator subordinated to a decomposition of the spatial domain and a standard spatial discretization. With this technique we obtain parallel algorithms which have the main advantages of classical domain decomposition methods and, besides, avoid iterative processes like Schwarz iterations, typical of them. The use of these embedded methods permits a fast variable step time integration process. [ABSTRACT FROM AUTHOR]

This paper presents a numerical scheme to approximate uɛ ∈ H10 (Ω), the weak solution of the problem [ABSTRACT FROM AUTHOR]

In this paper we consider an iterative substructuring method for solving system of equations arising from mortar Morley finite element discretization of a model fourth order elliptic problem in 2D. A parallel preconditioner for the interface problem is introduced using Additive Schwarz Method framework. The method is quasi-optimal i.e. the number of CG iterations for the preconditioned problem grows polylogarithmically as the sizes of the meshes decrease and it is independent of the jumps of the coefficients. [ABSTRACT FROM AUTHOR]

In this paper we study the condition number of the system resulting from C0 interior penalty methods for fourth order elliptic boundary value problems. We show that the condition number can be bounded by Ch-4 and that this bound is sharp, where h is the mesh size of the triangulation and C is a positive constant independent of the mesh size. [ABSTRACT FROM AUTHOR]

The paper deals with solving of contact problems with Coulomb friction for a system of 3D elastic bodies. The iterative method of successive approximations is used in order to find a fixed point of certain mapping that defines the solution. In each iterative step, an auxiliary problem with given friction is solved that is discretized by the FETI method. [ABSTRACT FROM AUTHOR]

We apply the finite element method to two-dimensional, incompressible MHD, using a streamfunction approach to enforce the divergence-free conditions on the magnetic and velocity fields. This problem was considered by Strauss and Longcope [1]. In this paper, we solve the problems with magnetic and velocity fields instead of the velocity stream function, magnetic flux, and their derivatives. Considering the multiscale nature of the tilt instability, we study the effect of domain resolution in the tilt instability problem. [ABSTRACT FROM AUTHOR]

We consider an efficient numerical method for computing time-harmonic acoustic scattering in a vertically layered media. One application for such problems is the detection of targets buried in a sediment. For this purpose it is useful to have a numerical approximation which can predict reasonably accurately the backscatter by such targets. In this paper, we study scattering by sound-soft targets when the interface between the water and sediment is wavy. Such problems are typically modeled using a Helmholtz equation with varying coefficients. [ABSTRACT FROM AUTHOR]

The parareal scheme (resp. PITA algorithm) proposed in [3] (resp. [2]) considers two levels of grids in time in order to split the domain in time-subdomains. A prediction of the solution is computed on the fine grid in parallel. Then at each interface between the time subdomains, the solution makes a jump between the previous initial boundary value (IBV) of the next time-subdomain . A correction of the IBV for the next fine grid iteration is then computed on the coarse grid in time. In this paper, we study adaptivity in the time slice decomposition based on an a posteriori numerical estimation obtained from the time step behavior on coarse grids. [ABSTRACT FROM AUTHOR]

Elliptic problems with multiscale coefficients have been studied to a great extent recently. Preconditioners based on standard domain decomposition methods often perform poorly when the variation of the coefficients inside the subdomains is large. In this paper we study the behaviour of domain decomposition methods based on linear coarsening for such problems and we also propose improved methods which use the notion of multiscale finite elements to define coarsening operators. [ABSTRACT FROM AUTHOR]

The objective of this paper is to present some numerical schemes for the time integration of parabolic problems that can recover from a failure of the computer system. We construct an algorithmic solution of the problem in the context of domain decomposition and distributed computing. [ABSTRACT FROM AUTHOR]

In this paper we present a domain decomposition method for the solution of a linear elliptic equation as a direct two-step procedure (in contrast to iterative-based Schwarz and Aitken-Schwarz procedures). First, local solutions to the nonhomogeneous equation are calculated on overlapping subdomains, and second, connection functions, which correct for mismatches on the subdomain intersections and solve the homogeneous equation in each subdomain, are obtained by solving a Cousin-like problem. [ABSTRACT FROM AUTHOR]

Quantization has diverse applications in many areas of science and engineering. In this paper, we present a new nonlinear multilevel algorithm that accelerates existing numerical methods for finding optimal quantizers. Both a theoretical framework for the convergence analysis and results of some computational experiments are provided. [ABSTRACT FROM AUTHOR]

Modal analysis of three-dimensional structures frequently involves finite element discretizations with millions of unknowns and requires computing hundreds or thousands of eigenpairs.We review in this paper methods based on domain decomposition for such eigenspace computations in structural dynamics. We distinguish approaches that solve the eigenproblem algebraically (with minimal connections to the underlying partial differential equation) from approaches that couple tightly the eigensolver with the partial differential equation. [ABSTRACT FROM AUTHOR]

BDDC (Balancing Domain Decomposition by Constraints) methods, so far developed for two levels [3, 7, 8], are similar to the balancing Neumann-Neumann algorithms. However, the BDDC coarse problem is given in terms of a set of primal constraints and the matrix of the coarse problem is generated and factored by direct solvers at the beginning of the computation. The coarse component of the preconditioner can ultimately become a bottleneck if the number of subdomains is very large. In this paper, two three-level BDDC methods are introduced for solving the coarse problem approximately in two and three dimensions, while still maintaining a good convergence rate. Estimates of the condition numbers are provided for the two three-level BDDC methods and numerical experiments are also discussed. [ABSTRACT FROM AUTHOR]

The Boundary Element Tearing and Interconnecting (BETI) methods have recently been introduced as boundary element counterparts of the well- established Finite Element Tearing and Interconnecting (FETI) methods. In this paper we present inexact data-sparse versions of the BETI methods which avoid the elimination of the primal unknowns and dense matrices. The data-sparse approximation of the matrices and the preconditioners involved is fully based on Fast Multipole Methods (FMM). This leads to robust solvers which are almost optimal with respect to the asymptotic complexity estimates. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to extend the BDDC (balancing domain decomposition by constraints) algorithm to saddle point problems that arise when mixed finite element methods are used to approximate the system of incompressible Stokes equations. The BDDC algorithms are defined in terms of a set of primal continuity constraints, which are enforced across the interface between the subdomains, and which provide a coarse space component of the preconditioner. Sets of such constraints are identified for which bounds on the rate of convergence can be established that are just as strong as previously known bounds for the elliptic case. The preconditioned operator is positive definite and a conjugate gradient method can be used. A close connection is also established between the BDDC and FETI-DP algorithms for the Stokes case. [ABSTRACT FROM AUTHOR]

We consider elliptic problems with discontinuous coefficients discretized by finite elements on non-matching triangulations across the interface using the mortar technique. The resulting discrete problem is solved by a FETI-DP method using a preconditioner with a special scaling described in a forthcoming paper by Dokeva, Dryja and Proskurowski. Experiments performed on up to a thousand processors show that this FETI-DP mortar method exhibits good parallel scalability. [ABSTRACT FROM AUTHOR]

The Overlapping Balancing Domain Decomposition (OBDD) methods can be considered as an extension of the Balancing Domain Decomposition (BDD) methods to the case of overlapping subdomains. This new approach, has been proposed and studied in [5, 3]. In this paper, we will discuss its practical parallel implementation and present numerical experiments on large unstructured meshes. [ABSTRACT FROM AUTHOR]

To solve electromagetic scattering problems in R3, the popular approach is to combine and couple finite and boundary elements. Common engineering practises in coupling finite and boundary elements usually result in non-symmetric and nonvariational formulations [5, 8]. The symmetric coupling between finite and boundary elements was first proposed by Costabel [2] in 1987. Since then, quite a few papers have been published on the topic of symmetric couplings. Among them, we list references [3, 4, 12, 7]. In particular, references [4, 12, 7] deal with variational formulations for solving electromagnetic wave radiation and scattering problems. Although the formulations detailed in [4, 12, 7] result in symmetric couplings between finite and boundary elements, they still suffer the notorious internal resonances. The purpose of this chapter is to present a variational formulation, which couples finite and boundary elements through non-conformal meshes. The formulation results in matrix equations that are symmetric, coercive, and free of internal resonances. [ABSTRACT FROM AUTHOR]

We study in this paper a model problem of advection diffusion type on a region which contains a subregion where it is sufficient to approximate the problem by the pure advection equation. We define coupling conditions at the interface between the two regions which lead to a coupled solution which approximates the fully viscous solution more accurately than other conditions from the literature, and we develop a fast algorithm to solve the coupled problem. [ABSTRACT FROM AUTHOR]

The automated multilevel sub-structuring (AMLS) method [2, 7, 3] is an extension of a simple sub-structuring method called component mode synthesis (CMS) [6, 4] originally developed in the 1960s. The recent work by Bennighof and Lehoucq [3] provides a high level mathematical description of the AMLS method in a continuous variational setting, as well as a framework for describing AMLS in matrix algebra notations. The AMLS approach has been successfully used in vibration and acoustic analysis of very large scale finite element models of automobile bodies [7]. In this paper, we evaluate the performance of AMLS on other types of applications. [ABSTRACT FROM AUTHOR]

This paper analyzes two-level Schwarz methods for matrices arising from the p-version finite element method on triangular and tetrahedral meshes. The coarse level consists of the lowest order finite element space. On the fine level, we investigate several decompositions with large or small overlap leading to optimal or close to optimal condition numbers. The analysis is confirmed by numerical experiments for a model problem. [ABSTRACT FROM AUTHOR]

This paper discusses the Schur complement viewpoint when developing parallel preconditioners for general sparse linear systems. Schur complement methods are pervasive in numerical linear algebra where they represent a canonical way of implementing divide-and-conquer principles. The goal of this note is to give a brief overview of recent progress made in using these techniques for solving general, irregularly structured, sparse linear systems. The emphasis is to point out the impact of Domain Decomposition ideas on the design of general purpose sparse system solution methods, as well as to show ideas that are of a purely algebraic nature. [ABSTRACT FROM AUTHOR]

The focus of this paper is a penalty-based strategy for preconditioning elliptic saddle point systems. As the starting point, we consider the regularization approach of Axelsson in which a related linear system, differing only in the (2,2) block of the coefficient matrix, is introduced. By choosing this block to be negative definite, the dual unknowns of the related system can be eliminated resulting in a positive definite primal Schur complement. Rather than solving the Schur complement system exactly, an approximate solution is obtained using a substructuring preconditioner. The approximate primal solution together with the recovered dual solution then define the preconditioned residual for the original system. [ABSTRACT FROM AUTHOR]

This paper presents an adaptive algebraic multigrid setup algorithm for positive definite linear systems arising from discretizations of elliptic partial differential equations. The proposed method uses compatible relaxation to select the set of coarse variables. The nonzero supports for the coarse-space basis are determined by approximation of the so-called two-level "ideal" interpolation operator. Then, an energy minimizing coarse basis is formed using an approach aimed to minimize the trace of the coarse-level operator. The variational multigrid solver resulting from the presented setup procedure is shown to be effective, without the need for parameter tuning, for some problems where current algorithms exhibit degraded performance. [ABSTRACT FROM AUTHOR]