<html><head></head><body><div style="color:#000; background-color:#fff; font-family:times new roman, new york, times, serif;font-size:16px">Dear Artur, <br><div id="yui_3_16_0_ym19_1_1468586733741_3241"><span></span></div><div id="yui_3_16_0_ym19_1_1468586733741_3249"><div id="yui_3_16_0_ym19_1_1468586733741_3405"> Out of a blend of curiosity and healthy naivity: have you tried complex shifted Laplace as <br></div><div id="yui_3_16_0_ym19_1_1468586733741_3352">a preconditioner? <br></div><div id="yui_3_16_0_ym19_1_1468586733741_4105"><br></div><div id="yui_3_16_0_ym19_1_1468586733741_4106"> Greetings, Domenico Lahaye.<br> </div></div><div id="yui_3_16_0_ym19_1_1468586733741_3244" class="qtdSeparateBR"><br></div><div style="display: block;" id="yui_3_16_0_ym19_1_1468586733741_3248" class="yahoo_quoted"> <div id="yui_3_16_0_ym19_1_1468586733741_3247" style="font-family: times new roman, new york, times, serif; font-size: 16px;"> <div id="yui_3_16_0_ym19_1_1468586733741_3246" style="font-family: HelveticaNeue, Helvetica Neue, Helvetica, Arial, Lucida Grande, sans-serif; font-size: 16px;"> <div id="yui_3_16_0_ym19_1_1468586733741_3245" dir="ltr"> <font id="yui_3_16_0_ym19_1_1468586733741_3253" face="Arial" size="2"> <hr id="yui_3_16_0_ym19_1_1468586733741_3404" size="1"> <b id="yui_3_16_0_ym19_1_1468586733741_3252"><span id="yui_3_16_0_ym19_1_1468586733741_3251" style="font-weight:bold;">From:</span></b> Sanjay Govindjee <s_g@berkeley.edu><br> <b><span style="font-weight: bold;">To:</span></b> petsc-users@mcs.anl.gov <br> <b id="yui_3_16_0_ym19_1_1468586733741_3467"><span id="yui_3_16_0_ym19_1_1468586733741_3466" style="font-weight: bold;">Sent:</span></b> Friday, July 15, 2016 11:02 AM<br> <b><span style="font-weight: bold;">Subject:</span></b> Re: [petsc-users] Multigrid with PML<br> </font> </div> <div id="yui_3_16_0_ym19_1_1468586733741_3354" class="y_msg_container"><br><div id="yiv9761378932"><div id="yui_3_16_0_ym19_1_1468586733741_3356">
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 clear="none">
<br clear="none">
</div><div id="yui_3_16_0_ym19_1_1468586733741_3358"><a id="yui_3_16_0_ym19_1_1468586733741_3357" rel="nofollow" shape="rect" target="_blank" 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 clear="none">
<br clear="none">
<a rel="nofollow" shape="rect" class="yiv9761378932moz-txt-link-freetext" target="_blank" href="http://dx.doi.org/10.1002/pamm.200700206">http://dx.doi.org/10.1002/pamm.200700206</a><br clear="none">
<br clear="none">
-sg<br clear="none">
<br clear="none">
<div class="yiv9761378932yqt5632725207" id="yiv9761378932yqt64645"><div id="yui_3_16_0_ym19_1_1468586733741_3359" class="yiv9761378932moz-cite-prefix">On 7/15/16 1:46 AM, Mark Adams wrote:<br clear="none">
</div>
<blockquote id="yui_3_16_0_ym19_1_1468586733741_3363" type="cite">
<div id="yui_3_16_0_ym19_1_1468586733741_3362" dir="ltr"><br clear="none">
<div id="yui_3_16_0_ym19_1_1468586733741_3364" class="yiv9761378932gmail_extra"><br clear="none">
<div id="yui_3_16_0_ym19_1_1468586733741_3366" class="yiv9761378932gmail_quote">On Thu, Jul 14, 2016 at 9:10 PM,
Barry Smith <span id="yui_3_16_0_ym19_1_1468586733741_3396" dir="ltr"><<a id="yui_3_16_0_ym19_1_1468586733741_3395" rel="nofollow" shape="rect" ymailto="mailto:bsmith@mcs.anl.gov" target="_blank" href="mailto:bsmith@mcs.anl.gov">bsmith@mcs.anl.gov</a>></span>
wrote:<br clear="none">
<blockquote id="yui_3_16_0_ym19_1_1468586733741_3365" class="yiv9761378932gmail_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 clear="none">
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 clear="none">
<br clear="none">
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 id="yui_3_16_0_ym19_1_1468586733741_3461" rel="nofollow" shape="rect" target="_blank" href="http://www.math.tau.ac.il/services/phd/dissertations/Singer_Ido.pdf">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 id="yui_3_16_0_ym19_1_1468586733741_3462" rel="nofollow" shape="rect" target="_blank" href="http://www.sciencedirect.com/science/article/pii/S0022247X11005063">http://www.sciencedirect.com/science/article/pii/S0022247X11005063</a><br clear="none">
<br clear="none">
<br clear="none">
Barry<br clear="none">
<br clear="none">
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 clear="none">
</blockquote>
<div><br clear="none">
</div>
<div id="yui_3_16_0_ym19_1_1468586733741_3397">Indefinite Helmholtz is hard unless you are not
shifting very far. This zero diagonals must come from PML.</div>
<div id="yui_3_16_0_ym19_1_1468586733741_3403"><br clear="none">
</div>
<div>First get rid of PML and see if you can solve anything
to your satisfaction.</div>
<div id="yui_3_16_0_ym19_1_1468586733741_3402"><br clear="none">
</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 id="yui_3_16_0_ym19_1_1468586733741_3399"> </div>
<blockquote class="yiv9761378932gmail_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="yiv9761378932"><br clear="none">
> On Jul 13, 2016, at 2:30 PM, Safin, Artur <<a rel="nofollow" shape="rect" ymailto="mailto:aks084000@utdallas.edu" target="_blank" href="mailto:aks084000@utdallas.edu">aks084000@utdallas.edu</a>>
wrote:<br clear="none">
><br clear="none">
</span>
<div>
<div class="yiv9761378932h5">> Dear PETSc community,<br clear="none">
><br clear="none">
> 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 clear="none">
><br clear="none">
> 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 clear="none">
><br clear="none">
> -ksp_type fgmres \<br clear="none">
> -pc_type gamg \<br clear="none">
> -mg_levels_pc_type bjacobi \<br clear="none">
> -pc_mg_type full \<br clear="none">
> -ksp_gmres_restart 150 \<br clear="none">
><br clear="none">
> Can anyone suggest some way of speeding up the
convergence? Any help would be appreciated. I am
attaching the output from kspview.<br clear="none">
><br clear="none">
> Best,<br clear="none">
><br clear="none">
> Artur<br clear="none">
><br clear="none">
</div>
</div>
> <kspview><br clear="none">
<br clear="none">
</blockquote>
</div>
<br clear="none">
</div>
</div>
</blockquote></div>
<br clear="none">
<pre class="yiv9761378932moz-signature">--
-----------------------------------------------
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</pre>
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