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@ARTICLE{Kleefeld:902317,
      author       = {Kleefeld, Andreas},
      title        = {{T}he hot spots conjecture can be false: {S}ome numerical
                      examples},
      journal      = {Advances in computational mathematics},
      volume       = {47},
      number       = {6},
      issn         = {1019-7168},
      address      = {Dordrecht [u.a.]},
      publisher    = {Springer Science + Business Media B.V},
      reportid     = {FZJ-2021-04174},
      pages        = {85},
      year         = {2021},
      abstract     = {The hot spots conjecture is only known to be true for
                      special geometries. This paper shows numerically that the
                      hot spots conjecture can fail to be true for easy to
                      construct bounded domains with one hole.The underlying
                      eigenvalue problem for the Laplace equation with Neumann
                      boundary condition is solved with boundary integral
                      equations yielding a non-linear eigenvalue problem. Its
                      discretization via the boundary element collocation method
                      in combination with the algorithm by Beyn yields highly
                      accurate results both for the first non-zero eigenvalue and
                      its corresponding eigenfunction which is due to
                      superconvergence. Additionally, it can be shown numerically
                      that the ratio between the maximal/minimal value inside the
                      domain and its maximal/minimal value on the boundary can be
                      larger than $1+10^{-3}$. Finally, numerical examples for
                      easy to construct domains with up to five holes are provided
                      which fail the hot spots conjecture as well.},
      cin          = {JSC},
      ddc          = {510},
      cid          = {I:(DE-Juel1)JSC-20090406},
      pnm          = {5112 - Cross-Domain Algorithms, Tools, Methods Labs (ATMLs)
                      and Research Groups (POF4-511)},
      pid          = {G:(DE-HGF)POF4-5112},
      typ          = {PUB:(DE-HGF)16},
      UT           = {WOS:000726273700001},
      doi          = {10.1007/s10444-021-09911-5},
      url          = {https://juser.fz-juelich.de/record/902317},
}