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  • 21
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    Monotone Multigrid Methods for Elliptic Variational Inequalities I. (1993)
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    Publication Date: 2014-02-26
    Description: Extending well--known linear concepts of successive subspace correction, we arrive at extended relaxation methods for elliptic variational inequalities. Extended underrelaxations are called monotone multigrid methods, if they are quasioptimal in a certain sense. By construction, all monotone multigrid methods are globally convergent. We take a closer look at two natural variants, which are called symmetric and unsymmetric multigrid methods, respectively. While the asymptotic convergence rates of the symmetric method suffer from insufficient coarse--grid transport, it turns out in our numerical experiments that reasonable application of the unsymmetric multigrid method may lead to the same efficiency as in the linear, unconstrained case.
    Keywords: ddc:000
    Language: English
    Type: reportzib , doc-type:preprint
    Format: application/postscript
    Format: application/pdf
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  • 22
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    Multilevel Methods for Elliptic Problems on Domains not Resolved by the Coarse Grid. (1993)
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    Publication Date: 2014-02-26
    Description: Elliptic boundary value problems are frequently posed on complicated domains which cannot be covered by a simple coarse initial grid as it is needed for multigrid like iterative methods. In the present article, this problem is resolved for selfadjoint second order problems and Dirichlet boundary conditions. The idea is to construct appropriate subspace decompositions of the corresponding finite element spaces by way of an embedding of the domain under consideration into a simpler domain like a square or a cube. Then the general theory of subspace correction methods can be applied.
    Keywords: ddc:000
    Language: English
    Type: reportzib , doc-type:preprint
    Format: application/postscript
    Format: application/pdf
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  • 23
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    Adaptive Multilevel-Methods in 3-Space Dimensions. (1992)
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    Publication Date: 2019-05-10
    Description: We consider the approximate solution of selfadjoint elliptic problems in three space dimensions by piecewise linear finite elements with respect to a highly non-uniform tetrahedral mesh which is generated adaptively. The arising linear systems are solved iteratively by the conjugate gradient method provided with a multilevel preconditioner. Here, the accuracy of the iterative solution is coupled with the discretization error. as the performance of hierarchical bases preconditioners deteriorate in three space dimensions, the BPX preconditioner is used, taking special care of an efficient implementation. Reliable a-posteriori estimates for the discretization error are derived from a local comparison with the approximation resulting from piecewise quadratic elements. To illustrate the theoretical results, we consider a familiar model problem involving reentrant corners and a real-life problem arising from hyperthermia, a recent clinical method for cancer therapy.
    Keywords: ddc:000
    Language: English
    Type: reportzib , doc-type:preprint
    Format: application/postscript
    Format: application/pdf
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  • 24
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    Monotone Multigrid Methods for Elliptic Variational Inequalities II. (1993)
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    Publication Date: 2014-02-26
    Description: We derive fast solvers for discrete elliptic variational inequalities of the second kind as resulting from the approximation by piecewise linear finite elements. Following the first part of this paper, monotone multigrid methods are considered as extended underrelaxations. Again, the coarse grid corrections are localized by suitable constraints, which in this case are fixed by fine grid smoothing. We consider the standard monotone multigrid method induced by the multilevel nodal basis and a truncated version. Global convergence results and asymptotic estimates for the convergence rates are given. The numerical results indicate a significant improvement in efficiency compared with previous multigrid approaches.
    Keywords: ddc:000
    Language: English
    Type: reportzib , doc-type:preprint
    Format: application/postscript
    Format: application/pdf
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  • 25
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    Globally Convergent Multigrid Methods for Porous Medium Type Problems (1997)
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    Publication Date: 2014-02-26
    Description: We consider the fast solution of large, piecewise smooth minimization problems as typically arising from the finite element discretization of porous media flow. For lack of smoothness, usual Newton multigrid methods cannot be applied. We propose a new approach based on a combination of convex minization with {\em constrained} Newton linearization. No regularization is involved. We show global convergence of the resulting monotone multigrid methods and give logarithmic upper bounds for the asymptotic convergence rates.
    Keywords: ddc:000
    Language: English
    Type: reportzib , doc-type:preprint
    Format: application/postscript
    Format: application/pdf
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  • 26
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    BOXES - a Program to Generate Triangulations from a Rectangular Domain Description. (1990)
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    Publication Date: 2014-02-26
    Description: BOXES computes a triangulation from a 2D domain description which consists of an arbitrary set of rectangles. Each rectangle may have attributes to control the triangulating process, define subdomain classes, or specify boundary conditions. The output of the program can be used as a coarse grid for KASKADE or one of its variants. Additional features are extensive checking of the user input, graphical display, and simple editing.
    Keywords: ddc:000
    Language: English
    Type: reportzib , doc-type:preprint
    Format: application/postscript
    Format: application/pdf
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  • 27
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    Mathematics cures virtual patients (2014)
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    Publication Date: 2019-01-29
    Language: English
    Type: incollection , doc-type:Other
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  • 28
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    Geometry processing (2014)
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    Publication Date: 2022-07-19
    Language: English
    Type: bookpart , doc-type:bookPart
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