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  <doc>
    <id>117</id>
    <completedYear/>
    <publishedYear/>
    <thesisYearAccepted/>
    <language>eng</language>
    <pageFirst/>
    <pageLast/>
    <pageNumber/>
    <edition/>
    <issue/>
    <volume/>
    <type>reportzib</type>
    <publisherName/>
    <publisherPlace/>
    <creatingCorporation/>
    <contributingCorporation/>
    <belongsToBibliography>0</belongsToBibliography>
    <completedDate>1993-09-01</completedDate>
    <publishedDate>1993-09-01</publishedDate>
    <thesisDateAccepted>--</thesisDateAccepted>
    <title language="eng">Multilevel Methods for Elliptic Problems on Domains not Resolved by the Coarse Grid.</title>
    <abstract language="eng">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.</abstract>
    <identifier type="serial">SC-93-21</identifier>
    <identifier type="opus3-id">116</identifier>
    <identifier type="urn">urn:nbn:de:0297-zib-1170</identifier>
    <enrichment key="SourceTitle">Appeared in: D. E. Keyes and J. Xu (eds.) Proc. of the 7th Int. Conf. on Domain Decomposition Methods 1993. Providence, RI: AMS 1994. Pp. 49-60</enrichment>
    <author>Ralf Kornhuber</author>
    <author>Harry Yserentant</author>
    <series>
      <title>ZIB-Report</title>
      <number>SC-93-21</number>
    </series>
    <collection role="ddc" number="000">Informatik, Informationswissenschaft, allgemeine Werke</collection>
    <collection role="institutes" number="">ZIB Allgemein</collection>
    <file>https://opus4.kobv.de/opus4-zib/files/117/SC-93-21.ps</file>
    <file>https://opus4.kobv.de/opus4-zib/files/117/SC-93-21.pdf</file>
  </doc>
  <doc>
    <id>13</id>
    <completedYear/>
    <publishedYear/>
    <thesisYearAccepted/>
    <language>eng</language>
    <pageFirst/>
    <pageLast/>
    <pageNumber/>
    <edition/>
    <issue/>
    <volume/>
    <type>reportzib</type>
    <publisherName/>
    <publisherPlace/>
    <creatingCorporation/>
    <contributingCorporation/>
    <belongsToBibliography>0</belongsToBibliography>
    <completedDate>1988-07-28</completedDate>
    <publishedDate>1988-07-28</publishedDate>
    <thesisDateAccepted>--</thesisDateAccepted>
    <title language="eng">Concepts of an Adaptive Hierarchical Finite Element Code.</title>
    <abstract language="eng">The paper presents the mathematical concepts underlying the new adaptive finite element code KASKADE, which, in its present form, applies to linear scalar second-order 2-D elliptic problems on general domains. Starting point for the new development is the recent work on hierarchical finite element bases due to Yserentant (1986). It is shown that this approach permits a flexible balance between iterative solver, local error estimator, and local mesh refinement device - which are the main components of an adaptive PDE code. Without use of standard multigrid techniques, the same kind of computational complexity is achieved - independent of any uniformity restrictions on the applied meshes. In addition, the method is extremely simple and all computations are purely local - making the method particularly attractive in view of parallel computing. The algorithmic approach is illustrated by a well-known critical test problem. {\bf Keywords:} finite elements, hierarchical basis, adaptive mesh refinement, preconditioned conjugate gradient methods.</abstract>
    <identifier type="serial">SC-88-05</identifier>
    <identifier type="opus3-id">13</identifier>
    <identifier type="urn">urn:nbn:de:0297-zib-131</identifier>
    <enrichment key="SourceTitle">Appeared in: IMPACT Comp. Sci. Eng. 1, pp. 3-35 (1989)</enrichment>
    <author>Peter Deuflhard</author>
    <author>P. Leinen</author>
    <author>Harry Yserentant</author>
    <series>
      <title>ZIB-Report</title>
      <number>SC-88-05</number>
    </series>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>finite elements</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>hierarchical basis</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>adaptive mesh refinement</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>preconditioned conjugate gradient methods</value>
    </subject>
    <collection role="ddc" number="000">Informatik, Informationswissenschaft, allgemeine Werke</collection>
    <collection role="msc" number="65F10">Iterative methods for linear systems [See also 65N22]</collection>
    <collection role="msc" number="65N20">Ill-posed problems</collection>
    <collection role="msc" number="65N30">Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods</collection>
    <collection role="msc" number="65N50">Mesh generation and refinement</collection>
    <collection role="institutes" number="">ZIB Allgemein</collection>
    <collection role="persons" number="deuflhard">Deuflhard, Peter</collection>
    <file>https://opus4.kobv.de/opus4-zib/files/13/SC-88-05.pdf</file>
  </doc>
  <doc>
    <id>71</id>
    <completedYear/>
    <publishedYear/>
    <thesisYearAccepted/>
    <language>eng</language>
    <pageFirst/>
    <pageLast/>
    <pageNumber/>
    <edition/>
    <issue/>
    <volume/>
    <type>reportzib</type>
    <publisherName/>
    <publisherPlace/>
    <creatingCorporation/>
    <contributingCorporation/>
    <belongsToBibliography>0</belongsToBibliography>
    <completedDate>1992-02-03</completedDate>
    <publishedDate>1992-02-03</publishedDate>
    <thesisDateAccepted>--</thesisDateAccepted>
    <title language="eng">A Basic Norm Equivalence for the Theory of Multilevel Methods.</title>
    <abstract language="eng">Subspace decompositions of finite element spaces based on $L2$-like orthogonal projections play an important role for the construction and analysis of multigrid like iterative methods. Recently several authors proved the equivalence of the associated discrete norms with the $H^1$-norm. The present report gives an elementary, self-contained derivation of this result which is based on the use of $ K$-functionals known from the theory of interpolation spaces. {\bf Keywords:} multilevel methods, nonuniform meshes, optimal convergence rates. {\bf AMS(MOS) Subject classifications:} 65N55, 65N30, 65N50.</abstract>
    <identifier type="serial">SC-92-01</identifier>
    <identifier type="opus3-id">70</identifier>
    <identifier type="doi">10.1007/BF01388699</identifier>
    <note>No preprint available</note>
    <enrichment key="SourceTitle">Appeared in: Numer. Math. 64 (1993) pp. 455-476</enrichment>
    <author>Folkmar A. Bornemann</author>
    <author>Harry Yserentant</author>
    <series>
      <title>ZIB-Report</title>
      <number>SC-92-01</number>
    </series>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>multilevel methods</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>nonuniform meshes</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>optimal convergence rates</value>
    </subject>
    <collection role="ddc" number="000">Informatik, Informationswissenschaft, allgemeine Werke</collection>
    <collection role="msc" number="65N30">Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods</collection>
    <collection role="msc" number="65N50">Mesh generation and refinement</collection>
    <collection role="msc" number="65N55">Multigrid methods; domain decomposition</collection>
    <collection role="institutes" number="">ZIB Allgemein</collection>
  </doc>
  <doc>
    <id>27</id>
    <completedYear/>
    <publishedYear/>
    <thesisYearAccepted/>
    <language>eng</language>
    <pageFirst/>
    <pageLast/>
    <pageNumber/>
    <edition/>
    <issue/>
    <volume/>
    <type>reportzib</type>
    <publisherName/>
    <publisherPlace/>
    <creatingCorporation/>
    <contributingCorporation/>
    <belongsToBibliography>0</belongsToBibliography>
    <completedDate>1989-11-17</completedDate>
    <publishedDate>1989-11-17</publishedDate>
    <thesisDateAccepted>--</thesisDateAccepted>
    <title language="eng">Two Preconditioners Based on the Multi-Level Splitting of Finite Element Spaces.</title>
    <abstract language="eng">The hierarchical basis preconditioner and the recent preconditioner of BRAMBLE, PASCIAK and XU are derived and analyzed within a joint framework. This discussion elucidates the close relationship between both methods. Special care is devoted to highly nonuniform meshes; our theory is based exclusively on local properties like the shape regularity of the finite elements.</abstract>
    <identifier type="serial">SC-89-09</identifier>
    <identifier type="opus3-id">27</identifier>
    <identifier type="urn">urn:nbn:de:0297-zib-274</identifier>
    <enrichment key="SourceTitle">Appeared in: Numer. Math. 58 (1990), pp. 163-184</enrichment>
    <author>Harry Yserentant</author>
    <series>
      <title>ZIB-Report</title>
      <number>SC-89-09</number>
    </series>
    <collection role="ddc" number="000">Informatik, Informationswissenschaft, allgemeine Werke</collection>
    <collection role="msc" number="65F10">Iterative methods for linear systems [See also 65N22]</collection>
    <collection role="msc" number="65F35">Matrix norms, conditioning, scaling [See also 15A12, 15A60]</collection>
    <collection role="msc" number="65N20">Ill-posed problems</collection>
    <collection role="msc" number="65N30">Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods</collection>
    <collection role="institutes" number="">ZIB Allgemein</collection>
    <file>https://opus4.kobv.de/opus4-zib/files/27/SC-89-09.pdf</file>
  </doc>
  <doc>
    <id>4</id>
    <completedYear/>
    <publishedYear/>
    <thesisYearAccepted/>
    <language>eng</language>
    <pageFirst/>
    <pageLast/>
    <pageNumber/>
    <edition/>
    <issue/>
    <volume/>
    <type>reportzib</type>
    <publisherName/>
    <publisherPlace/>
    <creatingCorporation/>
    <contributingCorporation/>
    <belongsToBibliography>0</belongsToBibliography>
    <completedDate>1987-04-01</completedDate>
    <publishedDate>1987-04-01</publishedDate>
    <thesisDateAccepted>--</thesisDateAccepted>
    <title language="eng">The Hierarchical Basis Multigrid Method.</title>
    <abstract language="eng">We derive and analyze the hierarchical basis-multigrid method for solving discretizations of self-adjoint, elliptic boundary value problems using piecewise linear triangular finite elements. The method is analyzed as a block symmetric Gauß- Seidel iteration with inner iterations, but it is strongly related to 2-level methods, to the standard multigrid V-cycle, and to earlier Jacobi-like hierarchical basis methods. The method is very robust, and has a nearly optimal convergence rate and work estimate. It is especially well suited to difficult problems with rough solutions, discretized using highly nonuniform, adaptively refined meshes.</abstract>
    <identifier type="serial">SC-87-02</identifier>
    <identifier type="opus3-id">4</identifier>
    <identifier type="urn">urn:nbn:de:0297-zib-44</identifier>
    <enrichment key="SourceTitle">Appeared in: Numer. Mathematik 52, p.427-458 (1988)</enrichment>
    <author>Randolph Bank</author>
    <author>Todd F. Dupont</author>
    <author>Harry Yserentant</author>
    <series>
      <title>ZIB-Report</title>
      <number>SC-87-02</number>
    </series>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>hierarchical basis</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>multigrid</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>finite elements</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>adaptive mesh refinement</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>preconditioned conjugate gradient methods</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>symmetric Gauss-Seidel</value>
    </subject>
    <collection role="ddc" number="000">Informatik, Informationswissenschaft, allgemeine Werke</collection>
    <collection role="msc" number="65F10">Iterative methods for linear systems [See also 65N22]</collection>
    <collection role="msc" number="65F35">Matrix norms, conditioning, scaling [See also 15A12, 15A60]</collection>
    <collection role="msc" number="65N20">Ill-posed problems</collection>
    <collection role="msc" number="65N30">Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods</collection>
    <collection role="institutes" number="">ZIB Allgemein</collection>
    <file>https://opus4.kobv.de/opus4-zib/files/4/SC-87-02.pdf</file>
  </doc>
  <doc>
    <id>8</id>
    <completedYear/>
    <publishedYear/>
    <thesisYearAccepted/>
    <language>eng</language>
    <pageFirst/>
    <pageLast/>
    <pageNumber/>
    <edition/>
    <issue/>
    <volume/>
    <type>reportzib</type>
    <publisherName/>
    <publisherPlace/>
    <creatingCorporation/>
    <contributingCorporation/>
    <belongsToBibliography>0</belongsToBibliography>
    <completedDate>1987-09-23</completedDate>
    <publishedDate>1987-09-23</publishedDate>
    <thesisDateAccepted>--</thesisDateAccepted>
    <title language="eng">Preconditioning Indefinite Discretization Matrices.</title>
    <abstract language="eng">The finite element discretization of many elliptic boundary value problems leads to linear systems with positive definite and symmetric coefficient matrices. Many efficient preconditioners are known for these systems. We show that these preconditioning matrices can be used also for the linear systems arising from boundary value problems which are potentially indefinite due to lower order terms in the partial differential equation. Our main tool is a careful algebraic analysis of the condition numbers and the spectra of perturbed matrices which are preconditioned by the same matrices as in the unperturbed case. {\bf Keywords: }Preconditioned conjugate gradient methods, finite elements. {\bf Subject Classification: } AMS(MOS):65F10, 65N20, 65N30.</abstract>
    <identifier type="serial">SC-87-06</identifier>
    <identifier type="opus3-id">8</identifier>
    <identifier type="urn">urn:nbn:de:0297-zib-81</identifier>
    <enrichment key="SourceTitle">Appeared in: Numer. Mathematik 54, p. 719-734 (1988)</enrichment>
    <author>Harry Yserentant</author>
    <series>
      <title>ZIB-Report</title>
      <number>SC-87-06</number>
    </series>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>preconditioned conjugate gradient methods</value>
    </subject>
    <subject>
      <language>eng</language>
      <type>uncontrolled</type>
      <value>finite elements</value>
    </subject>
    <collection role="ddc" number="000">Informatik, Informationswissenschaft, allgemeine Werke</collection>
    <collection role="msc" number="65F10">Iterative methods for linear systems [See also 65N22]</collection>
    <collection role="msc" number="65N20">Ill-posed problems</collection>
    <collection role="msc" number="65N30">Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods</collection>
    <collection role="institutes" number="">ZIB Allgemein</collection>
    <file>https://opus4.kobv.de/opus4-zib/files/8/SC-87-06.pdf</file>
  </doc>
  <doc>
    <id>5488</id>
    <completedYear/>
    <publishedYear>2014</publishedYear>
    <thesisYearAccepted/>
    <language>eng</language>
    <pageFirst>341</pageFirst>
    <pageLast>355</pageLast>
    <pageNumber/>
    <edition/>
    <issue/>
    <volume>1</volume>
    <type>bookpart</type>
    <publisherName>EMS Publishing House</publisherName>
    <publisherPlace/>
    <creatingCorporation/>
    <contributingCorporation/>
    <belongsToBibliography>0</belongsToBibliography>
    <completedDate>--</completedDate>
    <publishedDate>--</publishedDate>
    <thesisDateAccepted>--</thesisDateAccepted>
    <title language="eng">Geometry processing</title>
    <parentTitle language="eng">MATHEON - Mathematics for Key Technologies</parentTitle>
    <identifier type="isbn">978-3-03719-137-8</identifier>
    <identifier type="doi">10.4171/137</identifier>
    <enrichment key="Series">EMS Series in Industrial and Applied Mathematics</enrichment>
    <enrichment key="PeerReviewed">no</enrichment>
    <author>Konrad Polthier</author>
    <submitter>Christoph von Tycowicz</submitter>
    <author>Alexander Bobenko</author>
    <author>Klaus Hildebrandt</author>
    <author>Ralf Kornhuber</author>
    <author>Christoph von Tycowicz</author>
    <author>Harry Yserentant</author>
    <author>Günter M. Ziegler</author>
    <collection role="institutes" number="vis">Visual Data Analysis</collection>
    <collection role="persons" number="vontycowicz">Tycowicz, Christoph von</collection>
    <collection role="institutes" number="VDcC">Visual and Data-centric Computing</collection>
  </doc>
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