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  • 1
    Electronic Resource
    Electronic Resource
    Multilevel methods for mixed finite elements in three dimensions (1999)
    Springer
    Numerische Mathematik 82 (1999), S. 253-279 
    Details
    ISSN: 0945-3245
    Keywords: Mathematics Subject Classification (1991):65N30, 65N22, 65F10
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract. In this paper we consider second order scalar elliptic boundary value problems posed over three–dimensional domains and their discretization by means of mixed Raviart–Thomas finite elements [18]. This leads to saddle point problems featuring a discrete flux vector field as additional unknown. Following Ewing and Wang [26], the proposed solution procedure is based on splitting the flux into divergence free components and a remainder. It leads to a variational problem involving solenoidal Raviart–Thomas vector fields. A fast iterative solution method for this problem is presented. It exploits the representation of divergence free vector fields as ${\bf curl}$ s of the ${\bf H}({\bf curl})$ –conforming finite element functions introduced by Nédélec [43]. We show that a nodal multilevel splitting of these finite element spaces gives rise to an optimal preconditioner for the solenoidal variational problem: Duality techniques in quotient spaces and modern algebraic multigrid theory [50, 10, 31] are the main tools for the proof.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/s002110050419
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    Paper (German National Licenses)
    Fulltext
  • 2
    Electronic Resource
    Electronic Resource
    Multilevel Gauging for Edge Elements (2000)
    Springer
    Computing 64 (2000), S. 97-122 
    Details
    ISSN: 1436-5057
    Keywords: AMS Subject Classifications: 65N30, 65N55, 78A30. ; Key Words: Computational electromagnetism, edge elements, hierarchical bases, multilevel decomposition, vector potentials, gauging.
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science
    Notes: Abstract The vector potential of a solenoidal vector field, if it exists, is not unique in general. Any procedure that aims to determine such a vector potential typically involves a decision on how to fix it. This is referred to by the term gauging. Gauging is an important issue in computational electromagnetism, whenever discrete vector potentials have to be computed. In this paper a new gauging algorithm for discrete vector potentials is introduced that relies on a hierarchical multilevel decomposition. With minimum computational effort it yields vector potentials whose L 2-norm does not severely blow up. Thus the new approach compares favorably to the widely used co-tree gauging.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/s006070050005
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    Paper (German National Licenses)
    Fulltext
  • 3
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    Non-reflecting boundary conditions for Maxwell’s equations (2003)
    Details
    Publication Date: 2020-09-25
    Language: English
    Type: article , doc-type:article
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    OPUS
  • 4
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    Adaptive multilevel methods for edge element discretizations of Maxwell’s equations (1999)
    Details
    Publication Date: 2021-03-16
    Language: English
    Type: article , doc-type:article
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    OPUS
  • 5
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    Multilevel Solution of the Time-Harmonic Maxwell's Equations Based on Edge Elements (1996)
    Details
    Publication Date: 2014-02-26
    Description: A widely used approach for the computation of time-harmonic electromagnetic fields is based on the well-known double-curl equation for either $\vec E$ or $\vec H$. An appealing choice for finite element discretizations are edge elements, the lowest order variant of a $H(curl)$-conforming basis. However, the large nullspace of the curl-operator gives rise to serious drawbacks. It comprises a considerable part of all spectral modes on the finite element grid, polluting the solution with non-physical contributions and causing the deterioration of standard iterative solvers. We tackle these problems by a nested multilevel algorithm. After every V-cycle in the $H(curl)$-conforming basis, the non-physical contributions are removed by a projection scheme. It requires the solution of Poisson's equation in the nullspace, which can be carried out efficiently by another multilevel iteration. The whole procedure yields convergence rates independent of the refinement level of the mesh. Numerical examples demonstrate the efficiency of the method.
    Keywords: ddc:000
    Language: English
    Type: reportzib , doc-type:preprint
    Format: application/postscript
    Format: application/pdf
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  • 6
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    Adaptive Multilevel Methods for Edge Element Discretizations of Maxwell's Equations (1997)
    Details
    Publication Date: 2014-02-26
    Description: 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.
    Keywords: ddc:000
    Language: English
    Type: reportzib , doc-type:preprint
    Format: application/postscript
    Format: application/pdf
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    OPUS
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