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    I agree, this is an extra hard problem when you add PML to it.  Here
    is a link to a paper that presents a few tricks applied to some
    aspects of this problem.<br>
    <br>
    <meta http-equiv="content-type" content="text/html; charset=utf-8">
    <a href="http://dx.doi.org/10.1002/pamm.200700206">Koyama, T. and
      Govindjee, S., ``Solving generalized
      complex-symmetriceigenvalue problems arising fromresonant MEMS
      simulations
      with PETSc," in Proceedings in AppliedMathematics and Mechanics,
      1141701-1141702 (2008)</a>.<br>
    <br>
    <a class="moz-txt-link-freetext" href="http://dx.doi.org/10.1002/pamm.200700206">http://dx.doi.org/10.1002/pamm.200700206</a><br>
    <br>
    -sg<br>
    <br>
    <div class="moz-cite-prefix">On 7/15/16 1:46 AM, Mark Adams wrote:<br>
    </div>
    <blockquote
cite="mid:CADOhEh6bAfyDHrJB78wTuZ7qgKdgJTKn=eRDm8EFbUTW4R+3Kg@mail.gmail.com"
      type="cite">
      <div dir="ltr"><br>
        <div class="gmail_extra"><br>
          <div class="gmail_quote">On Thu, Jul 14, 2016 at 9:10 PM,
            Barry Smith <span dir="ltr"><<a moz-do-not-send="true"
                href="mailto:bsmith@mcs.anl.gov" target="_blank">bsmith@mcs.anl.gov</a>></span>
            wrote:<br>
            <blockquote class="gmail_quote" style="margin:0px 0px 0px
0.8ex;border-left-width:1px;border-left-style:solid;border-left-color:rgb(204,204,204);padding-left:1ex"><br>
                 This is a very difficult problem. I am not surprised
              that GAMG performs poorly, I would be surprised if it
              performed well at all.<br>
              <br>
                 I think you need to do some googling of   "helmholtz
              PML linear system solve" to find what other people have
              used. The first hit I got was this <a
                moz-do-not-send="true"
href="http://www.math.tau.ac.il/services/phd/dissertations/Singer_Ido.pdf"
                rel="noreferrer" target="_blank">http://www.math.tau.ac.il/services/phd/dissertations/Singer_Ido.pdf</a>
              and every iterative method he tried ended up requiring
              MANY iterations with refinement. This is 14 years old so
              there will be better suggestions out there. One that
              caught my eye was <a moz-do-not-send="true"
href="http://www.sciencedirect.com/science/article/pii/S0022247X11005063"
                rel="noreferrer" target="_blank">http://www.sciencedirect.com/science/article/pii/S0022247X11005063</a><br>
              <br>
              <br>
                Barry<br>
              <br>
              Just looking at the matrix makes it clear to me that
              conventional iterative methods are not going to work well,
              many of the diagonal entries are zero and even in rows
              with a diagonal entry it is much smaller in magnitude than
              the diagonal entries.<br>
            </blockquote>
            <div><br>
            </div>
            <div>Indefinite Helmholtz is hard unless you are not
              shifting very far. This zero diagonals must come from PML.</div>
            <div><br>
            </div>
            <div>First get rid of PML and see if you can solve anything
              to your satisfaction.</div>
            <div><br>
            </div>
            <div>I have a paper on this, using AMG, and I tried to be
              inclusive, but I did miss a potentially useful method of
              adding a complex shift to damp the system. You can Google
              something like 'complex shift helmholtz damp'.  If you are
              shifting deep (high frequency Helmholtz), then use direct
              solvers.</div>
            <div> </div>
            <blockquote class="gmail_quote" style="margin:0px 0px 0px
0.8ex;border-left-width:1px;border-left-style:solid;border-left-color:rgb(204,204,204);padding-left:1ex"><span
                class=""><br>
                > On Jul 13, 2016, at 2:30 PM, Safin, Artur <<a
                  moz-do-not-send="true"
                  href="mailto:aks084000@utdallas.edu">aks084000@utdallas.edu</a>>
                wrote:<br>
                ><br>
              </span>
              <div>
                <div class="h5">> Dear PETSc community,<br>
                  ><br>
                  > I am working on solving a Helmholtz problem with
                  PML. The issue is that I am finding it very hard to
                  deal with the resulting matrix system; I can get the
                  correct solution for coarse meshes, but it takes
                  roughly 2-4 times as long to converge for each
                  successively refined mesh. I've noticed that without
                  PML, I do not have problems with convergence speed.<br>
                  ><br>
                  > I am using the GMRES solver with GAMG as the
                  preconditioner (with block-Jacobi preconditioner for
                  the multigrid solves). I have also tried to assemble a
                  separate preconditioning matrix with the complex shift
                  1+0.5i, that does not seem to improve the results.
                  Currently I am running with<br>
                  ><br>
                  >    -ksp_type fgmres \<br>
                  >    -pc_type gamg \<br>
                  >    -mg_levels_pc_type bjacobi \<br>
                  >    -pc_mg_type full \<br>
                  >    -ksp_gmres_restart 150 \<br>
                  ><br>
                  > Can anyone suggest some way of speeding up the
                  convergence? Any help would be appreciated. I am
                  attaching the output from kspview.<br>
                  ><br>
                  > Best,<br>
                  ><br>
                  > Artur<br>
                  ><br>
                </div>
              </div>
              > <kspview><br>
              <br>
            </blockquote>
          </div>
          <br>
        </div>
      </div>
    </blockquote>
    <br>
    <pre class="moz-signature" cols="72">-- 
-----------------------------------------------
Sanjay Govindjee, PhD, PE
Professor of Civil Engineering

779 Davis Hall
University of California
Berkeley, CA 94720-1710

Voice:  +1 510 642 6060
FAX:    +1 510 643 5264
<a class="moz-txt-link-abbreviated" href="mailto:s_g@berkeley.edu">s_g@berkeley.edu</a>
<a class="moz-txt-link-freetext" href="http://www.ce.berkeley.edu/~sanjay">http://www.ce.berkeley.edu/~sanjay</a>
-----------------------------------------------

Books:  

Engineering Mechanics of Deformable 
Solids: A Presentation with Exercises
<a class="moz-txt-link-freetext" href="http://www.oup.com/us/catalog/general/subject/Physics/MaterialsScience/?view=usa&ci=9780199651641">http://www.oup.com/us/catalog/general/subject/Physics/MaterialsScience/?view=usa&ci=9780199651641</a>
<a class="moz-txt-link-freetext" href="http://ukcatalogue.oup.com/product/9780199651641.do">http://ukcatalogue.oup.com/product/9780199651641.do</a>
<a class="moz-txt-link-freetext" href="http://amzn.com/0199651647">http://amzn.com/0199651647</a>

Engineering Mechanics 3 (Dynamics) 2nd Edition
<a class="moz-txt-link-freetext" href="http://www.springer.com/978-3-642-53711-0">http://www.springer.com/978-3-642-53711-0</a>
<a class="moz-txt-link-freetext" href="http://amzn.com/3642537111">http://amzn.com/3642537111</a>

Engineering Mechanics 3, Supplementary Problems: Dynamics 
<a class="moz-txt-link-freetext" href="http://www.amzn.com/B00SOXN8JU">http://www.amzn.com/B00SOXN8JU</a>

-----------------------------------------------
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