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Domain Decomposition and Multilevel Techniques for Preconditioning OperatorsNepomnyaschikh, S. V. 30 October 1998 (has links)
Introduction In recent years, domain decomposition methods have been used extensively to efficiently solve boundary value problems for partial differential equations in complex{form domains. On the other hand, multilevel techniques on hierarchical data structures also have developed into an effective tool for the construction and analysis of fast solvers. But direct realization of multilevel techniques on a parallel computer system for the global problem in the original domain involves difficult communication problems. I this paper, we present and analyze a combination of these two approaches: domain decomposition and multilevel decomposition on hierarchical structures to design optimal preconditioning operators.

32 
SPCPM Po 3D  Users ManualApel, Th. 30 October 1998 (has links)
The experimental program ¨SPCPM Po 3D¨ is part of the ongoing research of the Chemnitz research group Scientific Parallel Computing (SPC) into finite element methods for problems over three dimensional domains. The package in its version 2.0 is documented in two manuals. The User's Manual provides an overview over the program, its capabilities, its installation, and handling. Moreover, test examples are explained. The aim of the Programmer's Manual is to provide a description of the algorithms and their realization. It is written for those who are interested in a deeper insight into the code, for example for improving and extending. In Version 2.0 the program can solve the Poisson equation and the Lam\'e system of linear elasticity with in general mixed boundary conditions of Dirichlet and Neumann type. The domain $\Omega\subset\R^3$ can be an arbitrarily bounded polyhedron. The input is a coarse mesh, a description of the data and some control parameters. The program distributes the elements of the coarse mesh to the processors, refines the elements, generates the system of equations using linear or quadratic shape functions, solves this system and offers graphical tools to display the solution. Further, the behavior of the algorithms can be monitored: arithmetic and communication time is measured, the discretization error is measured, different preconditioners can be compared. We plan to extend the program in the next future by including a multigrid solver, an error estimator and adaptive mesh refinement, as well as the treatment of coupled thermoelastic problems. The program has been developed for MIMD computers; it has been tested on Parsytec machines (GCPowerPlus128 with Motorola Power PC601 processors and GCel192 on transputer basis) and on workstation clusters using PVM. The special case of only one processor is included, that means the package can be compiled for single processor machines without any change in the source files.

33 
On the preconditioning in the domain decomposition technique for the pversion finite element method. Part IIvanov, S. A., Korneev, V. G. 30 October 1998 (has links)
Abstract Pversion finite element method for the second order elliptic equation in an arbitrary sufficiently smooth domain is studied in the frame of DD method. Two types square reference elements are used with the products of the integrated Legendre's polynomials for the coordinate functions. There are considered the estimates for the condition numbers, preconditioning of the problems arising on subdomains and the Schur complement, the derivation of the DD preconditioner. For the result we obtain the DD preconditioner to which corresponds the generalized condition number of order (logp )2 . The paper consists of two parts. In part I there are given some preliminary re sults for 1D case, condition number estimates and some inequalities for 2D reference element.

34 
On the preconditioning in the domain decomposition technique for the pversion finite element method. Part IIIvanov, S. A., Korneev, V. G. 30 October 1998 (has links)
Pversion finite element method for the second order elliptic equation in an arbitrary sufficiently smooth domain is studied in the frame of DD method. Two types square reference elements are used with the products of the integrated Legendre's polynomials for the coordinate functions. There are considered the estimates for the condition numbers, preconditioning of the problems arising on subdomains and the Schur complement, the derivation of the DD preconditioner. For the result we obtain the DD preconditioner to which corresponds the generalized condition number of order (logp )2 . The paper consists of two parts. In part I there are given some preliminary results for 1D case, condition number estimates and some inequalities for 2D reference element. Part II is devoted to the derivation of the Schur complement preconditioner and conditionality number estimates for the pversion finite element matrixes. Also DD preconditioning is considered.

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Zur Berechnung von Spannungs und Deformationsfeldern an InterfaceEcken im nichtlinearen Deformationsbereich auf ParallelrechnernScherzer, M., Meyer, A. 30 October 1998 (has links)
Using material models on the basis of the flow theory of plasticity the asymptotic behaviour of solid mechanics solutions in crack tips, interface corners etc. strongly depends on the local realized load trajectory. For incrementally proportional load paths the equations determining the asymptotic fields are very simple ones. The paper considers twodimensional statements in the neighbourhood of an interface corner consisting of two material ranges. At a distance from the corner the finite element nodes of a regular net are established in a polar coordinate system together with the displacement degrees of freedom. The main idea of the presented singular and nonsingular stress and deformation field calculation at interface corners characterizes an replacement of the corner neighbourhood effect to the surrounding body by introducing stiffness actions which in usual manner can be assembled together with the other element stiffness matrices to the global stiffness matrix of the body. According to this there exists an in teresting invariant stiffness independence in corner and crack neighbourhoods. The applied technique allows extensions to nonproportional local load increments simplifying the mathematical calculations for the presentation of stress and strain fields in this general case. All computations are made on modern parallel computers. Concrete examples show the advantages of the presented approach.

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Scalability, efficiency, and robustness of parallel multilevel solvers for nonlinear equationsHeise, B., Jung, M. 30 October 1998 (has links)
In this paper we compare the performance, scalability, and robustness of different parallel algorithms for the numerical solution of nonlinear boundary value problems arising in the magnetic field computation and in solid mechanics. These problems are discretized by using the finite element method with triangular meshes and piecewise linear functions. The nonlinearity is handled by a nested Newton solver, and the linear systems of algebraic equations within each Newton step are solved by means of various iterative solvers, namely multigrid methods and conjugate gradient methods with preconditioners based on domain decomposition, multigrid, or BPX techniques, respectively. The basis of the implementation of all solvers is a nonoverlapping domain decomposition data structure such that they are wellsuited for parallel machines with MIMD architecture.

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Partitionierung von FiniteElementeNetzenReichel, U. 30 October 1998 (has links)
The realization of the finite element method on parallel computers
is usually based on a domain decomposition approach. This paper is
concerned with the problem of finding an optimal decomposition and an
appropriate mapping of the subdomains to the processors. The quality
of this partitioning is measured in several metrics but it is also
expressed in the computing time for solving specific systems of
finite element equations.
The software environment is first described. In particular, the data
structure and the accumulation algorithm are introduced. Then
several partitioning algorithms are compared. Spectral bisection was
used with different modifications including KernighanLin refinement,
postprocessing techniques and terminal propagation. The final
recommendations should give good decompositions for all finite
element codes which are based on principles similar to ours.
The paper is a shortened English version of Preprint SFB393/9618
(Uwe Reichel: Partitionierung von FiniteElementeNetzen), SFB 393,
TU ChemnitzZwickau, December 1996. To be selfcontained, some material
of Preprint SPC95_5 (see below) is included. The paper appeared as
Preprint SFB393/9618a, SFB 393, TU ChemnitzZwickau, January 1997.

38 
Randkonzentrierte und adaptive hpFEMEibner, Tino 19 June 2006 (has links)
Die vorliegende Arbeit befasst sich mit verschiedenen Aspekten der hpFEM.
Insbesondere werden hierbei folgende Punkte genauer untersucht:
1. Das effiziente Aufstellen der Steifigkeitsmatrix auf Referenzelementen, die keine Tensorproduktstruktur besitzen.
2. Eine lokale Konvergenzbetrachtung für die randkonzentrierte hpFEM.
3. Ein MultilevelLöser für die randkonzentrierte hpFEM.
4. Eine Strategie für hpAdaptivität.

39 
Multiplication operators and its illposedness propertiesG.Fleischer 30 October 1998 (has links)
This paper deals with the characterization of multiplication operators,
especially with its behavior in the illposed case.
We want to classify the different types and degrees of illposedness. We give
some connections between this classification and regularization methods.

40 
Convergence of Asynchronous JacobiNewtonIterationsSchrader, U. 30 October 1998 (has links)
Asynchronous iterations often converge under different conditions than their syn chronous counterparts. In this paper we will study the global convergence of Jacobi Newtonlike methods for nonlinear equationsF x = 0. It is a known fact, that the synchronous algorithm converges monotonically, ifF is a convex Mfunction and the starting valuesx0 andy0 meet the conditionF x04 04F y0 . In the paper it will be shown, which modifications are necessary to guarantee a similar convergence behavior for an asynchronous computation.

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