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  • 1
    Publikationsdatum: 2014-02-26
    Beschreibung: Induction heating of large steel slabs can be described by a coupled system of nonlinear evolution equations of Stefan type representing the temporal and spatial distribution of the induced magnetic field and the generated temperature within the slab. Discretizing these equations implicitly in time and by finite differences in space, at each time step the solution of a system of difference inclusions is required. For the solution of that system two multi-grid algorithms are given which combined with a nested iteration type continuation strategy to proceed in time result in computationally highly efficient schemes for the numerical simulation of the induction heating process. {\bf Keywords:} induction heating, system of two coupled Stefan equations, multi-grid algorithms. {\bf Subject Classification:} AMS(MOS): 35K60, 35R35, 65H10, 65N05, 65N20, 78A25, 78A55.
    Schlagwort(e): ddc:000
    Sprache: Englisch
    Materialart: reportzib , doc-type:preprint
    Format: application/pdf
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  • 2
    Publikationsdatum: 2014-02-26
    Beschreibung: The solidification of an $ N $-component alloy is described by an initial boundary value problem for a system of degenerate parabolic equations modelling heat conduction and mass diffusion. Discretizing implicitly in time and by piecewise linear finite elements in the space variables, at each time step the solution of a system of quasivariational inequalities is required. For the numerical solution of that system, a multi-grid algorithm is developed by making use of game theoretic concepts and duality arguments from convex analysis. Finally, the efficiency of the algorithm is demonstrated by displaying numerical results for a ternary alloy.
    Schlagwort(e): ddc:000
    Sprache: Englisch
    Materialart: reportzib , doc-type:preprint
    Format: application/pdf
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  • 3
    Publikationsdatum: 2020-10-02
    Beschreibung: We consider the discretization of obstacle problems for second order elliptic differential operators in three space dimensions by piecewise linear finite elements. Linearizing the discrete problems by suitable active set strategies, the resulting linear sub--problems are solved iteratively by preconditioned cg--iterations. We propose a variant of the BPX preconditioner and prove an $O(j)$ estimate for the resulting condition number. To allow for local mesh refinement we derive semi--local and local a posteriori error estimates. The theoretical results are illustrated by numerical computations.
    Schlagwort(e): ddc:000
    Sprache: Englisch
    Materialart: reportzib , doc-type:preprint
    Format: application/postscript
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  • 4
    Publikationsdatum: 2021-03-16
    Sprache: Englisch
    Materialart: article , doc-type:article
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  • 5
    Publikationsdatum: 2014-02-26
    Beschreibung: We consider such variational inequalities which either describe obstacle problems or result from an implicit time discretization of moving boundary problems of two phase Stefan type. Based on a discretization in space by means of continuous, piecewise linear finite elements with respect to a nested hierarchy of triangulations, in both cases we use iterative processes consisting of inner and outer iterations. The outer iterations are either active set strategies or generalized Newton methods while the inner iterations are preconditioned cg- iterations with multilevel preconditioners.
    Schlagwort(e): ddc:000
    Sprache: Englisch
    Materialart: reportzib , doc-type:preprint
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  • 6
    Publikationsdatum: 2014-02-26
    Beschreibung: We consider the discretization of obstacle problems for the Laplacian by piecewise linear finite elements. Assuming that the discrete problems are reduced to a sequence of linear problems by suitable active set strategies, the linear problems are solved iteratively by preconditioned c-g iterations. The proposed preconditioners are treated theoretically as abstract additive Schwarz methods and are implemented as truncated hierarchical basis preconditioners. To allow for local mesh refinement we derive semi-local and local a posteriori error estimates, providing lower and upper estimates for the discretization error. The theoretical results are illustrated by numerical computations.
    Schlagwort(e): ddc:000
    Sprache: Englisch
    Materialart: reportzib , doc-type:preprint
    Format: application/postscript
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  • 7
    Publikationsdatum: 2014-02-26
    Beschreibung: The focus of this paper is on the efficient solution of boundary value problems involving the double-- curl operator. Those arise in the computation of electromagnetic fields in various settings, for instance when solving the electric or magnetic wave equation with implicit timestepping, when tackling time--harmonic problems or in the context of eddy--current computations. Their discretization is based on on N\'ed\'elec's {\bf H(curl}; $\Omega$)--conforming edge elements on unstructured grids. In order to capture local effects and to guarantee a prescribed accuracy of the approximate solution adaptive refinement of the grid controlled by a posteriori error estimators is employed. The hierarchy of meshes created through adaptive refinement forms the foundation for the fast iterative solution of the resulting linear systems by a multigrid method. The guiding principle underlying the design of both the error estimators and the multigrid method is the separate treatment of the kernel of the curl--operator and its orthogonal complement. Only on the latter we have proper ellipticity of the problem. Yet, exploiting the existence of computationally available discrete potentials for edge element spaces, we can switch to an elliptic problem in potential space to deal with nullspace of curl. Thus both cases become amenable to strategies of error estimation and multigrid solution developed for second order elliptic problems. The efficacy of the approach is confirmed by numerical experiments which cover several model problems and an application to waveguide simulation.
    Schlagwort(e): ddc:000
    Sprache: Englisch
    Materialart: reportzib , doc-type:preprint
    Format: application/postscript
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  • 8
    Digitale Medien
    Digitale Medien
    Springer
    Numerische Mathematik 36 (1981), S. 389-403 
    ISSN: 0945-3245
    Schlagwort(e): AMS(MOS): 65N 30 ; CR: 5.17
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary The paper deals with nonconforming finite element methods for the approximate solution of the interior boundary value problem for Maxwell equations in the time-harmonic case. The methods are based on penalization in the boundary conditions of total reflexion. Qualitative convergence results are obtained by a-priori estimates which are proven in the first part of this paper. The main object is to establish estimates for the global discretization error in various norms of the underlying spaces of approximating vector fields.
    Materialart: Digitale Medien
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  • 9
    Digitale Medien
    Digitale Medien
    Springer
    Numerische Mathematik 49 (1986), S. 239-254 
    ISSN: 0945-3245
    Schlagwort(e): AMS(MOS): 65N20 ; CR: G1.8
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary In this paper we develop multi-grid algorithms for the numerical solution of Hamilton-Jacobi-Bellman equations. The proposed schemes result from a combination of standard multi-grid techniques and the iterative methods used by Lions and mercier in [11]. A convergence result is given and the efficiency of the algorithms is illustrated by some numerical examples.
    Materialart: Digitale Medien
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  • 10
    Digitale Medien
    Digitale Medien
    Springer
    Annali di matematica pura ed applicata 136 (1984), S. 183-212 
    ISSN: 1618-1891
    Quelle: Springer Online Journal Archives 1860-2000
    Thema: Mathematik
    Notizen: Summary Using basic techniques from the theory of interpolation spaces equivalence theorems are established for the intermediate spaces between a given Banach space A and the domain D(Λr) of the r-th power of the infinitesimal generator Λ of a strongly continuous cosine operator function C. The results are applied to the study of second order evolution equations including regularity, order reduction and approximation by finite difference methods.
    Materialart: Digitale Medien
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