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  • 1
    Electronic Resource
    Electronic Resource
    Springer
    Numerische Mathematik 49 (1986), S. 11-37 
    ISSN: 0945-3245
    Keywords: AMS(MOS):65N300 ; 65B05 ; CR: G1.8
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary The elliptic Ritz projection with linear finite elements is shown to admit asymptotic error expansions on certain uniform meshes. This justifies the application of Richardson extrapolation for increasing the accuracy.
    Type of Medium: Electronic Resource
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  • 2
    Electronic Resource
    Electronic Resource
    Springer
    Numerische Mathematik 52 (1987), S. 539-564 
    ISSN: 0945-3245
    Keywords: AMS(OMS): 65N30, 65B05, 73K25 ; CR: G1.8
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary Standard finite element schemes applied to boundary value problems on domains with reentrant corners suffer from a global loss of accuracy caused by the influence of the corner singularities. For a model situation, it is shown that this pollution effect can be completely described in terms of an asymptotic error expansion with respect to fractional powers of the mesh size parameter. This provides the basis of an extrapolation procedure which recovers the full order of accuracy for the solution as well as for the corresponding stress intensity factors.
    Type of Medium: Electronic Resource
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  • 3
    Electronic Resource
    Electronic Resource
    Springer
    Computing and visualization in science 2 (1999), S. 107-122 
    ISSN: 1433-0369
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract. In this paper, we describe recent developments in the design and implementation of Navier-Stokes solvers based on finite element discretization. The most important ingredients are residual driven a posteriori mesh refinement, fully coupled defect-correction iteration for linearization, and optimal multigrid preconditioning. These techniques were systematically developed for computing incompressible viscous flows in general domains. Recently they have been extended to compressible low-Mach flows involving chemical reactions. The potential of automatic mesh adaptation together with multilevel techniques is illustrated by several examples, (1) the accurate prediction of drag and lift coefficients, (2) the determination of CARS-signals of species concentration in flow reactors, (3) the computation of laminar flames.
    Type of Medium: Electronic Resource
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  • 4
    Electronic Resource
    Electronic Resource
    Springer
    Computing 55 (1995), S. 271-288 
    ISSN: 1436-5057
    Keywords: 65F10 ; Finite elements ; multigrid methods ; error control
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science
    Description / Table of Contents: Zusammenfassung Wir behandeln das Problem einer adaptiven Fehlerkontrolle bei Finite-Elemente-Methoden unter Enschluß des Fehlers, der durch ungenaue Lösung der diskreten Gelichungen entsteht. Wir beweisen A-posteriori-Fehlerabschätzungen für ein elliptisches Modellproblem, welches mit linearen finiten-Elementen diskretisiert wird. Die diskreten Gleichungen werden mit Hilfe des kanonischen Finite-Elemente-Mehrgitterverfahrens gelöst. Die Beweise beruhen auf der Kombination der «starken” stabilitätseigenschaft des zugrundeliegenden Differentialoperators und der Galerkin-Orthogonalität sowohl des Finite-Elemente-als auch des Mehrgitterverfahrens.
    Notes: Abstract We consider the problem of adaptive error control in the finite element method including the error resulting from, inexact solution of the discrete equations. We prove a posteriori error estimates for a prototype elliptic model problem discretized by the finite element with a canomical multigrid algorithm. The proofs are based on a combination of so-called strong stability and, the orthogonality inherent in both the finite element method can the multigrid algorithm.
    Type of Medium: Electronic Resource
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  • 5
    Electronic Resource
    Electronic Resource
    Springer
    Computing 49 (1992), S. 11-23 
    ISSN: 1436-5057
    Keywords: 65M55 ; 65Y05 (primary) ; 65M60 (secondary) ; Domain decomposition ; parallel time stepping ; parabolic problems ; transputer system
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science
    Description / Table of Contents: Zusammenfassung Die Implementierung von Lösungs-methoden für parabolische Probleme erfordert eine ausreichende Balance zwischen der parallelen Effizienz voll-expliziter Schemata und der Notwendigkeit von Stabilität und Genauigkeit, welche einen gewissen Grad an Implizitheit bedingt. Als ein Kompromiß wird ein Gebietszerlegungsverfahren vorgeschlagen, welches lokal implizit ist auf leicht überlappenden Teilgebieten, die lokalen Randdaten aber durch einen einfachen expliziten Prozeß fortpflanzt. Die Analyse dieses Algorithmus zeigt, daß er zufriedenstellende Stabilitäts- und Approximationseigenschaften besitzt und effektiv parallelisiert werden kann. Diese theoretischen Resultate werden bestätigt durch numerische Tests auf einem Transputer-System.
    Notes: Abstract In the parallel implementation of solution methods for parabolic problems one has to find a proper balance between the parallel efficiency of a fully explicit scheme and the need for stability and accuracy which requires some degree of implicitness. As a compromise a domain splitting scheme is proposed which is locally implicit on slightly overlapping subdomains but propagates the corresponding boundary data by a simple explicit process. The analysis of this algorithm shows that it has satisfactory stability and approximation properties and can be effectively parallelized. These theoretical results are confirmed by numerical tests on a transputer system.
    Type of Medium: Electronic Resource
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  • 6
    Electronic Resource
    Electronic Resource
    Springer
    Computational mechanics 6 (1990), S. 221-236 
    ISSN: 1432-0924
    Keywords: 65 N 30 ; 73 K 10 ; 73 K 25
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mechanical Engineering, Materials Science, Production Engineering, Mining and Metallurgy, Traffic Engineering, Precision Mechanics
    Notes: Summary Most of the existing convergence theory of mixed finite element methods for solving the plate bending problem converns the model case of a purely clamped or simply supported plate with sufficiently regular boundary. The extension of this analysis to more complicated situations encounters two major difficulties: first, the problem of verifying the stability of the schemes in the case of a partially free boundary and, second, the reduction of the solution's regularity in the presence of reentrant corners or changes in the type of the boundary conditions. In this paper these questions are studied for the approximation of the Kirchhoff plate model by one of the mixed finite element schemes due to L. R. Herrmann, the so-called “first Herrmann scheme”. It is shown that this method converges on any polygonal domain and for all usual boundary conditions. The proof is based on the fact that this particular mixed scheme is algebraically equivalent to a nonconforming displacement method.
    Type of Medium: Electronic Resource
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  • 7
    Electronic Resource
    Electronic Resource
    Springer
    Computational mechanics 19 (1997), S. 434-446 
    ISSN: 1432-0924
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mechanical Engineering, Materials Science, Production Engineering, Mining and Metallurgy, Traffic Engineering, Precision Mechanics
    Notes: Abstract Recently a refined approach to error control in finite element (FE) discretisations has been proposed, Becker and Rannacher (1995b), (1996), which uses weighted a posteriori error estimates derived via duality arguments. The conventional strategies for mesh refinement in FE models of problems from elasticity theory are mostly based on a posteriori error estimates in the energy norm. Such estimates reflect the approximation properties of the finite element ansatz by local interpolation constants while the stability properties of the continuous model enter through a global coercivity constant. However, meshes generated on the basis of such global error estimates are not appropriate in cases where the domain consists of very heterogeneous materials and for the computation of local quantities, e.g., point values or contour integrals. This deficiency is cured by using certain local norms of the dual solution directly as weights multiplying the local residuals of the computed solution. In general, these weights have to be evaluated numerically in the course of the refinement process, yielding almost optimal meshes for various kinds of error measures. This feed-back approach is developed here for primal as well as mixed FE discretisations of the fundamental problem in linear elasticity.
    Type of Medium: Electronic Resource
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  • 8
    Electronic Resource
    Electronic Resource
    Springer
    Computational mechanics 21 (1998), S. 123-133 
    ISSN: 1432-0924
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mechanical Engineering, Materials Science, Production Engineering, Mining and Metallurgy, Traffic Engineering, Precision Mechanics
    Notes: Abstract In this paper a new technique for a posteriori error control and adaptive mesh design is presented for finite element models in perfect plasticity. The approach is based on weighted a posteriori error estimates derived by duality arguments as proposed in Becker and Rannacher (1996) and Rannacher and Suttmeier (1997) for linear problems. The conventional strategies for mesh refinement in finite element methods are mostly based on a posteriori error estimates for the global energy norm in terms of local residuals of the computed solution. These estimates reflect the approximation properties of the trial functions by local interpolation constants while the stability property of the continuous model enters through a global coercivity constant. However, meshes generated on the basis of such global error estimates are not appropriate in computing local quantities as point values or contour integrals and in the case of nonlinear material behavior. More accurate and efficient error estimation can be achieved by using suitable weights which can be obtained numerically in the course of the refinement process from the solutions of linearized dual problems. This feed-back approach is developed here for primal-mixed finite element models in linear-elastic perfect plasticity.
    Type of Medium: Electronic Resource
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  • 9
    Electronic Resource
    Electronic Resource
    Chichester : Wiley-Blackwell
    International Journal for Numerical Methods in Fluids 22 (1996), S. 325-352 
    ISSN: 0271-2091
    Keywords: Navier-Stokes equations ; artificial boundary conditions ; flux and pressure conditions ; finite elements ; Engineering ; Engineering General
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mechanical Engineering, Materials Science, Production Engineering, Mining and Metallurgy, Traffic Engineering, Precision Mechanics
    Notes: Fluid dynamical problems are often conceptualized in unbounded domains. However, most methods of numerical simulation then require a truncation of the conceptual domain to a bounded one, thereby introducing artificial boundaries. Here we analyse our experience in choosing artificial boundary conditions implicitly through the choice of variational formulations. We deal particularly with a class of problems that involve the prescription of pressure drops and/or net flux conditions.
    Additional Material: 19 Ill.
    Type of Medium: Electronic Resource
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  • 10
    Electronic Resource
    Electronic Resource
    Chichester : Wiley-Blackwell
    International Journal for Numerical Methods in Fluids 28 (1998), S. 1-22 
    ISSN: 0271-2091
    Keywords: accretion disk ; compressible Navier-Stokes equations ; stratified flow ; operator splitting ; hydrodyamic code ; boundary layer ; Engineering ; Numerical Methods and Modeling
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mechanical Engineering, Materials Science, Production Engineering, Mining and Metallurgy, Traffic Engineering, Precision Mechanics
    Notes: A new numerical approach based on consistent operator splitting is presented for computing compressible, highly stratified flows in astrophysics. The algorithm is particularly designed to search for steady or almost steady solutions for the time-dependent Navier-Stokes equations, describing viscous flow under the influence of a strong gravitational field. The algorithm proposed is multidimensional and works in Cartesian, cylindrical or spherical co-ordinates. It uses a second-order finite volume scheme with third-order upwinding and a second-order time discretization. An adaptive time step control and monotonic multilevel grid distribution has been incorporated to speed up convergence. This method has been incorporated into a hydrodynamical code by which, for the first time, for two-dimensional models the dynamics of the boundary layer in the accretion disk around a compact star could be computed over the whole viscous time scale. © 1998 John Wiley & Sons, Ltd.
    Additional Material: 18 Ill.
    Type of Medium: Electronic Resource
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