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  • ddc:000  (18)
  • 1
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    Multi-Grid Solution of Two Coupled Stefan Equations Arising in Induction Heating of Large Steel Slabs. (1988)
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    Publication Date: 2014-02-26
    Description: 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.
    Keywords: ddc:000
    Language: English
    Type: reportzib , doc-type:preprint
    Format: application/pdf
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  • 2
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    Self Adaptive Computation of the Breakdown Voltage of Planar pn-Junctions with Multistep Field Plates. (1991)
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    Publication Date: 2014-02-26
    Description: The breakdown voltage highly depends on the electric field in the depletion area whose computation is the most time consuming part of the simulation. We present a self adaptive Finite Element Method which reduces dramatically the required computation time compared to usual Finite Difference Methods. A numerical example illustrates the efficiency and reliability of the algorithm.
    Keywords: ddc:000
    Language: English
    Type: reportzib , doc-type:preprint
    Format: application/pdf
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  • 3
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    Adaptive Multilevel-Methods for Obstacle Problems in Three Space Dimensions. (1993)
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    Publication Date: 2020-10-02
    Description: 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.
    Keywords: ddc:000
    Language: English
    Type: reportzib , doc-type:preprint
    Format: application/postscript
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  • 4
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    Monotone Iterations for Elliptic Variational Inequalities (1998)
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    Publication Date: 2014-02-26
    Description: A wide range of free boundary problems occurring in engineering andindustry can be rewritten as a minimization problem for astrictly convex, piecewise smooth but non--differentiable energy functional.The fast solution of related discretized problemsis a very delicate question, because usual Newton techniquescannot be applied. We propose a new approach based on convex minimization and constrained Newton type linearization. While convex minimization provides global convergence of the overall iteration, the subsequent constrained Newton type linearization is intended to accelerate the convergence speed. We present a general convergence theory and discuss several applications.
    Keywords: ddc:000
    Language: English
    Type: reportzib , doc-type:preprint
    Format: application/postscript
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  • 5
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    Self Adaptive FEM Simulation of Reverse Biased pn-Junctions. (1990)
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    Publication Date: 2014-02-26
    Description: The potential distribution of reverse biased pn-junctions can be described by a double obstacle problem for the Laplacian. This problem is solved by a self adaptive Finite Element Method involving automatic termination criteria for the iterative solver, local error estimation and local mesh refinement. Special attention is paid to the efficient resolution of the geometries typically arising in semiconductor device simulation. The algorithm is applied to a reverse biased pn- junction with multi-step field plate and stop- electrode to illustrate its efficiency and reliability.
    Keywords: ddc:000
    Language: English
    Type: reportzib , doc-type:preprint
    Format: application/pdf
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  • 6
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    On Robust Multigrid Methods for Piecewise Smooth Variational Problems (1996)
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    Publication Date: 2014-02-26
    Description: We consider the fast solution of large, piecewise smooth minimization problems as resulting from the approximation of elliptic free boundary problems. The most delicate question in constructing a multigrid method for a nonlinear, non--smooth problem is how to represent the nonlinearity on the coarse grids. This process usually involves some kind of linearization. The basic idea of monotone multigrid methods to be presented here is first to select a neighborhood of the actual smoothed iterate in which a linearization is possible and then to constrain the coarse grid correction to this neighborhood. Such a local linearization allows to control the local corrections at each coarse grid node in such a way that the energy functional is monotonically decreasing. This approach leads to globally convergent schemes which are robust with respect to local singularities of the given problem. The numerical performance is illustrated by approximating the well-known Barenblatt solution of the porous medium equation.
    Keywords: ddc:000
    Language: English
    Type: reportzib , doc-type:preprint
    Format: application/postscript
    Format: application/pdf
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  • 7
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    Multilevel Preconditioned CG-Iterations for Variational Inequalities. (1991)
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    Publication Date: 2014-02-26
    Description: 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.
    Keywords: ddc:000
    Language: English
    Type: reportzib , doc-type:preprint
    Format: application/pdf
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  • 8
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    Discretization and Iterative Solution of Convection Diffusion Equations. (1992)
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    Publication Date: 2021-03-16
    Description: We propose an extended box method which turns out to be a variant of standard finite element methods in the case of pure diffusion and an extension of backward differencing to irregular grids if only convective transport is present. Together with the adaptive orientation proposed in a recent paper and a streamline ordering of the unknowns, this discretization leads to a highly efficient adaptive method for the approximation of internal layers in the case of large local Peclet numbers.
    Keywords: ddc:000
    Language: English
    Type: reportzib , doc-type:preprint
    Format: application/pdf
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  • 9
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    Adaptive Multilevel - Methods for Obstacle Problems. (1991)
    Details
    Publication Date: 2014-02-26
    Description: 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.
    Keywords: ddc:000
    Language: English
    Type: reportzib , doc-type:preprint
    Format: application/postscript
    Format: application/pdf
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  • 10
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    A Posteriori Error Estimates for Elliptic Problems. (1993)
    Details
    Publication Date: 2019-05-10
    Description: {\def\enorm {\mathop{\mbox{\boldmath{$|\!|$}}}\nolimits} Let $u \in H$ be the exact solution of a given self--adjoint elliptic boundary value problem, which is approximated by some $\tilde{u} \in {\cal S}$, $\cal S$ being a suitable finite element space. Efficient and reliable a posteriori estimates of the error $\enorm u - \tilde{u}\enorm $, measuring the (local) quality of $\tilde{u}$, play a crucial role in termination criteria and in the adaptive refinement of the underlying mesh. A well--known class of error estimates can be derived systematically by localizing the discretized defect problem using domain decomposition techniques. In the present paper, we provide a guideline for the theoretical analysis of such error estimates. We further clarify the relation to other concepts. Our analysis leads to new error estimates, which are specially suited to three space dimensions. The theoretical results are illustrated by numerical computations.}
    Keywords: ddc:000
    Language: English
    Type: reportzib , doc-type:preprint
    Format: application/postscript
    Format: application/pdf
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