% IMPORTANT: The following is UTF-8 encoded.  This means that in the presence
% of non-ASCII characters, it will not work with BibTeX 0.99 or older.
% Instead, you should use an up-to-date BibTeX implementation like “bibtex8” or
% “biber”.

@ARTICLE{Bolten:848186,
      author       = {Bolten, M. and Rittich, Hannah},
      title        = {{F}ourier {A}nalysis of {P}eriodic {S}tencils in
                      {M}ultigrid {M}ethods},
      journal      = {SIAM journal on scientific computing},
      volume       = {40},
      number       = {3},
      issn         = {1095-7197},
      address      = {Philadelphia, Pa.},
      publisher    = {SIAM},
      reportid     = {FZJ-2018-03452},
      pages        = {A1642-A1668},
      year         = {2018},
      abstract     = {Many applications require the numerical solution of a
                      partial differential equation (PDE), leading to large and
                      sparse linear systems. Often a multigrid method can solve
                      these systems efficiently. To adapt a multigrid method to a
                      given problem, local Fourier analysis (LFA) can be used. It
                      provides quantitative predictions about the behavior of the
                      components of a multigrid method. In this paper we
                      generalize LFA to handle what we call periodic stencils. An
                      operator given by a periodic stencil has a block Fourier
                      symbol representation. It gives a way to compute the
                      spectral radius and norm of the operator. Furthermore block
                      Fourier symbols can be used to find out how an operator acts
                      on smooth/oscillatory input and whether its output will be
                      smooth/oscillatory. This information can then be used to
                      construct efficient smoothers and coarse grid corrections.
                      We consider a particular PDE with jumping coefficients and
                      show that it leads to a periodic stencil. LFA shows that the
                      Jacobi method is a suitable smoother for this problem and an
                      operator dependent interpolation is better than linear
                      interpolation, as suggested by numerical experiments
                      described in the literature. If an operator is given by an
                      ordinary stencil, then block smoothers yield periodic
                      stencils if the blocks correspond to rectangles in the
                      domain. LFA shows that the block Jacobi and the red-black
                      block Jacobi method efficiently reduce more frequencies than
                      their pointwise versions. Further, it yields that a block
                      smoother used in combination with aggressive coarsening can
                      to some degree compensate for the reduced convergence rate
                      caused by aggressive coarsening.},
      cin          = {JSC},
      ddc          = {004},
      cid          = {I:(DE-Juel1)JSC-20090406},
      pnm          = {511 - Computational Science and Mathematical Methods
                      (POF3-511) / SPPEXA - Software for Exascale Computing
                      (214420555)},
      pid          = {G:(DE-HGF)POF3-511 / G:(GEPRIS)214420555},
      typ          = {PUB:(DE-HGF)16},
      UT           = {WOS:000436986000039},
      doi          = {10.1137/16M1073959},
      url          = {https://juser.fz-juelich.de/record/848186},
}