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@PHDTHESIS{Theisen:988413,
      author       = {Theisen, Lambert},
      othercontributors = {Stamm, Benjamin and Reusken, Arnold and Henning, Patrick},
      title        = {{S}calable domain decomposition eigensolvers for
                      {S}chrödinger operators in anisotropic structures},
      school       = {RWTH Aachen University},
      type         = {Dissertation},
      address      = {Aachen},
      publisher    = {RWTH Aachen University},
      reportid     = {RWTH-2024-06163},
      pages        = {1 Online-Ressource : Illustrationen},
      year         = {2024},
      note         = {Veröffentlicht auf dem Publikationsserver der RWTH Aachen
                      University; Dissertation, RWTH Aachen University, 2024},
      abstract     = {This thesis presents the construction and analysis of
                      scalable preconditioning strategies for the linear
                      Schrödinger eigenvalue problem with periodic potentials in
                      anisotropic structures. As only some dimensions of the
                      computational domain expand to infinity, the resulting
                      eigenvalue gap between the first and second eigenvalue
                      vanishes, posing a significant challenge for iterative
                      solvers. For these iterative eigenvalue solvers, we provide
                      a quasi-optimal shift-and-invert preconditioning strategy
                      such that the iterative eigenvalue algorithms converge in
                      constant iterations for different domain sizes. In its
                      analysis, we derive an analytic factorization of the
                      eigenpairs and use directional homogenization to analyze the
                      asymptotic behavior. The resulting easy-to-calculated unit
                      cell problem can be used within a shift-and-invert
                      preconditioning strategy. This approach leads to a uniformly
                      bounded number of eigensolver iterations. Numerical examples
                      illustrate the effectiveness of this quasi-optimal
                      preconditioning strategy if direct solvers are used since
                      the shifting strategy, by definition, leads to a smaller
                      eigenvalue for the resulting shifted operator, which, in
                      turn, results in a high condition number. We also provide a
                      two-level domain decomposition preconditioner for iterative
                      linear solvers to overcome this issue. As the calculation of
                      the quasi-optimal shift already offered the solution to a
                      spectral cell problem as limiting eigenfunction, it is
                      logical to use it as a generator to construct a coarse
                      space. Indeed, it is the case that the resulting two-level
                      additive Schwarz preconditioner is independent of the
                      domain's anisotropy since we obtain a condition number bound
                      using the theory of spectral coarse spaces despite the need
                      for only one basis function per subdomain for the coarse
                      solver. We provide several numerical examples illustrating
                      the effectiveness of both methods separately and combine
                      them in the end to show their combined scalability.},
      cin          = {111710 / 110000},
      ddc          = {510},
      cid          = {$I:(DE-82)111710_20140620$ / $I:(DE-82)110000_20140620$},
      typ          = {PUB:(DE-HGF)11},
      doi          = {10.18154/RWTH-2024-06163},
      url          = {https://publications.rwth-aachen.de/record/988413},
}