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Dear Mathew,</div>
<div style="font-family: Calibri, Arial, Helvetica, sans-serif; font-size: 12pt; color: rgb(0, 0, 0);">
thank you for your response. I double checked and snes is converged in all steps. following is output of -snes_converged_reason</div>
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<br>
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0 TS dt 0.0001 time 0.
<div>copy!</div>
<div>Write output at step= 0!</div>
<div> 0 SNES Function norm 1.465357113711e+01</div>
<div> 1 SNES Function norm 1.131764004094e+02</div>
<div> Nonlinear solve converged due to CONVERGED_ITS iterations 1</div>
<div> 0 SNES Function norm 5.657793773845e+01</div>
<div> 1 SNES Function norm 9.539766295097e-02</div>
<div> Nonlinear solve converged due to CONVERGED_ITS iterations 1</div>
<div>1 TS dt 0.0002 time 0.0001</div>
<div>copy!</div>
<div> 0 SNES Function norm 1.734774181586e+02</div>
<div> 1 SNES Function norm 1.009139331913e+00</div>
<div> Nonlinear solve converged due to CONVERGED_ITS iterations 1</div>
<div>2 TS dt 0.000324496 time 0.0003</div>
<div>copy!</div>
<div> 0 SNES Function norm 1.200543689889e+01</div>
<div> 1 SNES Function norm 1.715000763566e-02</div>
<div> Nonlinear solve converged due to CONVERGED_ITS iterations 1</div>
<div>3 TS dt 0.000471924 time 0.000624496</div>
<div>copy!</div>
<div> 0 SNES Function norm 8.374168676869e+00</div>
<div> 1 SNES Function norm 3.806213226152e-03</div>
<div> Nonlinear solve converged due to CONVERGED_ITS iterations 1</div>
<div>4 TS dt 0.00076859 time 0.00109642</div>
<div>copy!</div>
<div> 0 SNES Function norm 9.316857431814e+00</div>
<div> 1 SNES Function norm 3.125732297714e-02</div>
<div> Nonlinear solve converged due to CONVERGED_ITS iterations 1</div>
<div>5 TS dt 0.00153718 time 0.00186501</div>
<div>copy!</div>
<div> 0 SNES Function norm 9.820005256073e+00</div>
<div> 1 SNES Function norm 8.107044589689e-04</div>
<div> Nonlinear solve converged due to CONVERGED_ITS iterations 1</div>
<div>6 TS dt 0.00218053 time 0.00340219</div>
<div>copy!</div>
<div> 0 SNES Function norm 5.122834112454e+00</div>
<div> 1 SNES Function norm 1.662895139952e-03</div>
<div> Nonlinear solve converged due to CONVERGED_ITS iterations 1</div>
<div>7 TS dt 0.0022609 time 0.00558272</div>
<div>copy!</div>
<div> 0 SNES Function norm 1.423878996243e+00</div>
<div> 1 SNES Function norm 1.329199025944e-03</div>
<div> Nonlinear solve converged due to CONVERGED_ITS iterations 1</div>
<div>8 TS dt 0.0045218 time 0.00784362</div>
<div>copy!</div>
<div> 0 SNES Function norm 1.241101191899e+00</div>
<div> 1 SNES Function norm 1.758309073907e-03</div>
<div> Nonlinear solve converged due to CONVERGED_ITS iterations 1</div>
<div> 0 SNES Function norm 2.174982578512e+00</div>
<div> 1 SNES Function norm 1.977451891592e-03</div>
<div> Nonlinear solve converged due to CONVERGED_ITS iterations 1</div>
<div>9 TS dt 0.00309088 time 0.0111379</div>
<div>copy!</div>
<div> 0 SNES Function norm 7.952246195856e-01</div>
<div> 1 SNES Function norm 3.315360582766e-04</div>
<div> Nonlinear solve converged due to CONVERGED_ITS iterations 1</div>
<div>10 TS dt 0.00302529 time 0.0142288</div>
<div>copy!</div>
<div> 0 SNES Function norm 7.852078318680e-01</div>
<div> 1 SNES Function norm 6.819399397977e-04</div>
<div> Nonlinear solve converged due to CONVERGED_ITS iterations 1</div>
<div>11 TS dt 0.0036818 time 0.0172541</div>
<div>copy!</div>
<div> 0 SNES Function norm 1.269332622839e+00</div>
<div> 1 SNES Function norm 1.508985874154e-03</div>
<div> Nonlinear solve converged due to CONVERGED_ITS iterations 1</div>
<div>12 TS dt 0.00736361 time 0.0209359</div>
<div>copy!</div>
<div> 0 SNES Function norm 6.143748829896e+00</div>
<div> 1 SNES Function norm 2.172457806998e-01</div>
<div> Nonlinear solve converged due to CONVERGED_ITS iterations 1</div>
<div>13 TS dt 0.00792431 time 0.0282995</div>
<div>copy!</div>
<div> 0 SNES Function norm 2.762542847702e+01</div>
<div> 1 SNES Function norm 1.329294167855e+01</div>
<div> Nonlinear solve converged due to CONVERGED_ITS iterations 1</div>
<div> 0 SNES Function norm 2.287537833706e+01</div>
<div> 1 SNES Function norm 1.608258731248e+00</div>
<div> Nonlinear solve converged due to CONVERGED_ITS iterations 1</div>
<div> 0 SNES Function norm 1.279894262237e+01</div>
<div> 1 SNES Function norm 5.112278399366e-02</div>
<div> Nonlinear solve converged due to CONVERGED_ITS iterations 1</div>
<div>14 TS dt 0.0017228 time 0.0301754</div>
<div>copy!</div>
<div> 0 SNES Function norm 4.714428103053e+00</div>
<div> 1 SNES Function norm 3.141482420272e-03</div>
<div> Nonlinear solve converged due to CONVERGED_ITS iterations 1</div>
<div>15 TS dt 0.00174124 time 0.0318982</div>
<div>copy!</div>
<div> 0 SNES Function norm 5.161696766061e+00</div>
<div> 1 SNES Function norm 3.434924981822e-03</div>
<div> Nonlinear solve converged due to CONVERGED_ITS iterations 1</div>
<div> 0 SNES Function norm 1.440940181693e+01</div>
<div> 1 SNES Function norm 2.060277373865e-02</div>
<div> Nonlinear solve converged due to CONVERGED_ITS iterations 1</div>
<div>16 TS dt 0.00143688 time 0.0334492</div>
<div>copy!</div>
<div> 0 SNES Function norm 4.954103004576e+00</div>
<div> 1 SNES Function norm 1.705855165536e-03</div>
<div> Nonlinear solve converged due to CONVERGED_ITS iterations 1</div>
<div>17 TS dt 0.00287377 time 0.0348861</div>
<div>copy!</div>
<div> 0 SNES Function norm 1.757129021431e+01</div>
<div> 1 SNES Function norm 3.191948709005e-02</div>
<div> Nonlinear solve converged due to CONVERGED_ITS iterations 1</div>
<div> 0 SNES Function norm 3.031772340693e+01</div>
<div> 1 SNES Function norm 7.097653811013e-02</div>
<div> Nonlinear solve converged due to CONVERGED_ITS iterations 1</div>
<div>18 TS dt 0.00203648 time 0.0368402</div>
<div>copy!</div>
<div> 0 SNES Function norm 1.629074198276e+01</div>
<div> 1 SNES Function norm 9.774829708834e-03</div>
<div> Nonlinear solve converged due to CONVERGED_ITS iterations 1</div>
<div> 0 SNES Function norm 4.007821179267e+01</div>
<div> 1 SNES Function norm 4.322266335869e-02</div>
<div> Nonlinear solve converged due to CONVERGED_ITS iterations 1</div>
<div> 0 SNES Function norm 2.348161408350e+01</div>
<div> 1 SNES Function norm 5.218780461268e-03</div>
<div> Nonlinear solve converged due to CONVERGED_ITS iterations 1</div>
<div> 0 SNES Function norm 1.789799437816e+01</div>
<div> 1 SNES Function norm 2.727867493741e-03</div>
<div> Nonlinear solve converged due to CONVERGED_ITS iterations 1</div>
<div>19 TS dt 0.000223859 time 0.0370644</div>
<div>copy!</div>
<div> 0 SNES Function norm 1.164475864748e+00</div>
<div> 1 SNES Function norm 5.580165229404e-03</div>
<div> Nonlinear solve converged due to CONVERGED_ITS iterations 1</div>
<div>20 TS dt 0.000209603 time 0.0372882</div>
<div>copy!</div>
<div>Write output at step= 20!</div>
<div> 0 SNES Function norm 7.548512298179e-01</div>
<div> 1 SNES Function norm 8.191710667867e-03</div>
<div> Nonlinear solve converged due to CONVERGED_ITS iterations 1</div>
<div> 0 SNES Function norm 5.113158684224e+00</div>
<div> 1 SNES Function norm 4.059400705957e-03</div>
<div> Nonlinear solve converged due to CONVERGED_ITS iterations 1</div>
<div> 0 SNES Function norm 3.408310084009e+00</div>
<div> 1 SNES Function norm 5.257240981430e-03</div>
<div> Nonlinear solve converged due to CONVERGED_ITS iterations 1</div>
<div>21 TS dt 7.85133e-05 time 0.0373492</div>
<div>copy!</div>
<div> 0 SNES Function norm 4.156375747211e-01</div>
<div> 1 SNES Function norm 3.052885436956e-02</div>
<div> Nonlinear solve converged due to CONVERGED_ITS iterations 1</div>
<div>22 TS dt 9.58699e-05 time 0.0374277</div>
<div>copy!</div>
<div> 0 SNES Function norm 4.583956492983e-01</div>
<div> 1 SNES Function norm 1.176944126767e-01</div>
<div> Nonlinear solve converged due to CONVERGED_ITS iterations 1</div>
<div>23 TS dt 9.32603e-05 time 0.0375236</div>
<div>copy!</div>
<div> 0 SNES Function norm 4.969090229611e-01</div>
<div> 1 SNES Function norm 1.738678566558e-02</div>
<div> Nonlinear solve converged due to CONVERGED_ITS iterations 1</div>
<div>24 TS dt 0.000186521 time 0.0376169</div>
<div>copy!</div>
<div> 0 SNES Function norm 6.406466977155e-01</div>
<div> 1 SNES Function norm 8.183507106613e-03</div>
<div> Nonlinear solve converged due to CONVERGED_ITS iterations 1</div>
<div>25 TS dt 0.000213369 time 0.0378034</div>
<div>copy!</div>
<div> 0 SNES Function norm 6.235579283905e-01</div>
<div> 1 SNES Function norm 7.038633582283e-02</div>
<div> Nonlinear solve converged due to CONVERGED_ITS iterations 1</div>
<div>26 TS dt 0.000217545 time 0.0380168</div>
<div>copy!</div>
<div> 0 SNES Function norm 5.375653976237e-01</div>
<div> 1 SNES Function norm 7.318538883142e-03</div>
<div> Nonlinear solve converged due to CONVERGED_ITS iterations 1</div>
<div>27 TS dt 0.000384287 time 0.0382343</div>
<div>copy!</div>
<div> 0 SNES Function norm 8.638617026715e-01</div>
<div> 1 SNES Function norm 1.596171692523e-03</div>
<div> Nonlinear solve converged due to CONVERGED_ITS iterations 1</div>
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<div id="divRplyFwdMsg" dir="ltr"><font face="Calibri, sans-serif" style="font-size:11pt" color="#000000"><b>From:</b> Matthew Knepley <knepley@gmail.com><br>
<b>Sent:</b> Tuesday, April 20, 2021 7:23 AM<br>
<b>To:</b> Sepideh Kavousi <skavou1@lsu.edu><br>
<b>Cc:</b> petsc-users@mcs.anl.gov <petsc-users@mcs.anl.gov><br>
<b>Subject:</b> Re: [petsc-users] convergence slow</font>
<div> </div>
</div>
<div>
<div dir="ltr">
<div dir="ltr">On Mon, Apr 19, 2021 at 5:43 PM Sepideh Kavousi <<a href="mailto:skavou1@lsu.edu">skavou1@lsu.edu</a>> wrote:<br>
</div>
<div class="x_gmail_quote">
<blockquote class="x_gmail_quote" style="margin:0px 0px 0px 0.8ex; border-left:1px solid rgb(204,204,204); padding-left:1ex">
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<div style="font-family:Calibri,Arial,Helvetica,sans-serif; font-size:12pt; color:rgb(0,0,0)">
<span style="font-size:12pt; font-family:Calibri,Arial,Helvetica,sans-serif; font-weight:normal; text-indent:0px">Hello,</span><br>
<span style="font-size:12pt; font-family:Calibri,Arial,Helvetica,sans-serif; font-weight:normal; text-indent:0px">I want to solve PF solidification+Navier stokes using Finite different method. The Poisson equation is much more complicated that the one for just
a fluid-flow problems (because it includes different terms order parameter terms).</span></div>
<div style="font-family:Calibri,Arial,Helvetica,sans-serif; font-size:12pt; color:rgb(0,0,0)">
<span style="font-size:12pt; font-family:Calibri,Arial,Helvetica,sans-serif; font-weight:normal; text-indent:0px">The code converges but the convergence rate is very slow. I have tried different options. Following is the best options I found but the convergence
is still very slow. I did not include the ksp_convergence data because it takes around 1000 liner solver steps for each snes step.
<br>
</span></div>
<div style="font-family:Calibri,Arial,Helvetica,sans-serif; font-size:12pt; color:rgb(0,0,0)">
<span style="font-size:12pt; font-family:Calibri,Arial,Helvetica,sans-serif; font-weight:normal; text-indent:0px">I look in literature, but I did not find any data on PF+navierstokes solutions using Petsc. Do you have any suggestions how should I improve the
convergence?</span></div>
</div>
</blockquote>
<div><br>
</div>
<div>First question: Are your nonlinear solves converging?</div>
<div><br>
</div>
<div> I see you have -snes_type ksponly, but it looks like your SNES iteration _diverges_ sometimes, and mostly fails to converge.</div>
<div> Is this correct?</div>
<div><br>
</div>
<div> Thanks,</div>
<div><br>
</div>
<div> Matt</div>
<div> </div>
<blockquote class="x_gmail_quote" style="margin:0px 0px 0px 0.8ex; border-left:1px solid rgb(204,204,204); padding-left:1ex">
<div dir="ltr">
<div style="font-family:Calibri,Arial,Helvetica,sans-serif; font-size:12pt; color:rgb(0,0,0)">
<span style="font-size:12pt; font-family:Calibri,Arial,Helvetica,sans-serif; font-weight:normal; text-indent:0px">Best,</span></div>
<div style="font-family:Calibri,Arial,Helvetica,sans-serif; font-size:12pt; color:rgb(0,0,0)">
<span style="font-size:12pt; font-family:Calibri,Arial,Helvetica,sans-serif; font-weight:normal; text-indent:0px">Sepideh<br>
</span></div>
<div style="font-family:Calibri,Arial,Helvetica,sans-serif; font-size:12pt; color:rgb(0,0,0)">
<span style="font-size:12pt; font-family:Calibri,Arial,Helvetica,sans-serif; font-weight:normal; text-indent:0px"><span id="x_gmail-m_1424732035757749725🙂"></span></span><br>
</div>
<div style="font-family:Calibri,Arial,Helvetica,sans-serif; font-size:12pt; color:rgb(0,0,0)">
<br>
</div>
<div style="font-family:Calibri,Arial,Helvetica,sans-serif; font-size:12pt; color:rgb(0,0,0)">
ibrun -np 136 ./one.out -ts_monitor -snes_fd_color -ts_max_snes_failures -1 -ts_type bdf -ts_bdf_adapt -snes_linesearch_type l2 -snes_type ksponly -pc_type fieldsplit -pc_fieldsplit_0_fields 0,1 -pc_fieldsplit_1_fields 2,3 -pc_fieldsplit_2_fields 4 -pc_fieldsplit_type
multiplicative -fieldsplit_0_pc_type bjacobi -fieldsplit_1_pc_type bjacobi -fieldsplit_2_pc_type ilu -fieldsplit_0_ksp_type preonly -fieldsplit_1_ksp_type preonly -fieldsplit_2_ksp_type preonly -sub_pc_type ilu -sub_ksp_type preonly -snes_monitor -log_view
log.txt<br>
</div>
<div style="font-family:Calibri,Arial,Helvetica,sans-serif; font-size:12pt; color:rgb(0,0,0)">
<br>
</div>
<div style="font-family:Calibri,Arial,Helvetica,sans-serif; font-size:12pt; color:rgb(0,0,0)">
<div>TACC: Starting up job 7605521</div>
<div>TACC: Starting parallel tasks...</div>
<div>nu_nd=14948.511 !</div>
<div>tau/w=26225.459!</div>
<div>initial!</div>
<div>0 TS dt 0.0001 time 0.</div>
<div>copy!</div>
<div>Write output at step= 0!</div>
<div> 0 SNES Function norm 1.465357113711e+01</div>
<div> 1 SNES Function norm 1.131764004117e+02</div>
<div> 0 SNES Function norm 5.657793773935e+01</div>
<div> 1 SNES Function norm 9.539772669908e-02</div>
<div>1 TS dt 0.0002 time 0.0001</div>
<div>copy!</div>
<div> 0 SNES Function norm 1.734774194628e+02</div>
<div> 1 SNES Function norm 1.009134558763e+00</div>
<div>2 TS dt 0.000324128 time 0.0003</div>
<div>copy!</div>
<div> 0 SNES Function norm 1.199860231725e+01</div>
<div> 1 SNES Function norm 1.698885201785e-02</div>
<div>3 TS dt 0.000470679 time 0.000624128</div>
<div>copy!</div>
<div> 0 SNES Function norm 8.344934987594e+00</div>
<div> 1 SNES Function norm 4.082958856578e-03</div>
<div>4 TS dt 0.000769357 time 0.00109481</div>
<div>copy!</div>
<div> 0 SNES Function norm 9.338443428763e+00</div>
<div> 1 SNES Function norm 2.472903267511e-03</div>
<div>5 TS dt 0.00153871 time 0.00186416</div>
<div>copy!</div>
<div> 0 SNES Function norm 9.836384162921e+00</div>
<div> 1 SNES Function norm 6.383053421790e-03</div>
<div>6 TS dt 0.00215245 time 0.00340288</div>
<div>copy!</div>
<div> 0 SNES Function norm 5.043289760603e+00</div>
<div> 1 SNES Function norm 2.168332362305e-03</div>
<div>7 TS dt 0.0022353 time 0.00555533</div>
<div>copy!</div>
<div> 0 SNES Function norm 1.421255264272e+00</div>
<div> 1 SNES Function norm 1.750393691475e-03</div>
<div>8 TS dt 0.0044706 time 0.00779063</div>
<div>copy!</div>
<div> 0 SNES Function norm 1.216746389225e+00</div>
<div> 1 SNES Function norm 5.339123694240e-04</div>
<div> 0 SNES Function norm 2.168489818312e+00</div>
<div> 1 SNES Function norm 1.928809773415e-03</div>
<div>9 TS dt 0.00310217 time 0.0110885</div>
<div>copy!</div>
<div> 0 SNES Function norm 8.021629529051e-01</div>
<div> 1 SNES Function norm 2.161901383228e-04</div>
<div>10 TS dt 0.00300302 time 0.0141906</div>
<div>copy!</div>
<div> 0 SNES Function norm 7.731190774223e-01</div>
<div> 1 SNES Function norm 2.807020573793e-04</div>
<div>11 TS dt 0.00367968 time 0.0171937</div>
<div>copy!</div>
<div> 0 SNES Function norm 1.263830181443e+00</div>
<div> 1 SNES Function norm 1.049133009231e-02</div>
<div>12 TS dt 0.00704796 time 0.0208733</div>
<div>copy!</div>
<div> 0 SNES Function norm 5.619818748359e+00</div>
<div> 1 SNES Function norm 1.569218805659e-01</div>
<div>13 TS dt 0.00742462 time 0.0279213</div>
<div>copy!</div>
<div> 0 SNES Function norm 2.268051808662e+01</div>
<div> 1 SNES Function norm 7.055762930884e+00</div>
<div> 0 SNES Function norm 2.173806325395e+01</div>
<div> 1 SNES Function norm 1.564571270312e+00</div>
<div> 0 SNES Function norm 1.208038223084e+01</div>
<div> 1 SNES Function norm 5.192087605517e-02</div>
<div>14 TS dt 0.00186927 time 0.0299244</div>
<div>copy!</div>
<div> 0 SNES Function norm 4.797722218343e+00</div>
<div> 1 SNES Function norm 3.975928994156e-03</div>
<div>15 TS dt 0.00189622 time 0.0317937</div>
<div>copy!</div>
<div> 0 SNES Function norm 5.510229536082e+00</div>
<div> 1 SNES Function norm 4.496811351139e-03</div>
<div>16 TS dt 0.00172953 time 0.0336899</div>
<div>copy!</div>
<div> 0 SNES Function norm 6.945417639798e+00</div>
<div> 1 SNES Function norm 4.569326335543e-03</div>
<div>17 TS dt 0.00218527 time 0.0354194</div>
<div>copy!</div>
<div> 0 SNES Function norm 1.351474156680e+01</div>
<div> 1 SNES Function norm 1.719775600164e-02</div>
<div>18 TS dt 0.00224386 time 0.0376047</div>
<div>copy!</div>
<div> 0 SNES Function norm 2.174969556700e+01</div>
<div> 1 SNES Function norm 3.814212178889e-02</div>
<div> 0 SNES Function norm 4.891415655138e+01</div>
<div> 1 SNES Function norm 6.772042718901e-02</div>
<div> 0 SNES Function norm 2.828143585009e+01</div>
<div> 1 SNES Function norm 6.965842377755e-03</div>
<div> 0 SNES Function norm 2.168193335917e+01</div>
<div> 1 SNES Function norm 2.463212234975e-03</div>
<div>19 TS dt 0.000205482 time 0.0378175</div>
<div>copy!</div>
<div> 0 SNES Function norm 1.409435206457e+00</div>
<div> 1 SNES Function norm 1.859628755245e-02</div>
<div>20 TS dt 0.000188785 time 0.0380229</div>
<div>copy!</div>
<div>Write output at step= 20!</div>
<div> 0 SNES Function norm 5.900261363044e-01</div>
<div> 1 SNES Function norm 2.653576084845e-02</div>
<div> 0 SNES Function norm 5.122905363140e+00</div>
<div> 1 SNES Function norm 4.297068425492e-03</div>
<div> 0 SNES Function norm 3.428448792746e+00</div>
<div> 1 SNES Function norm 1.000610664096e-02</div>
<div>21 TS dt 6.50511e-05 time 0.0380738</div>
<div>copy!</div>
<div> 0 SNES Function norm 3.700405789863e-01</div>
<div> 1 SNES Function norm 6.263642369355e-02</div>
<div>22 TS dt 7.93452e-05 time 0.0381388</div>
<div>copy!</div>
<div> 0 SNES Function norm 5.282939288286e-01</div>
<div> 1 SNES Function norm 3.003209840754e-01</div>
<div>23 TS dt 7.50242e-05 time 0.0382181</div>
<div>copy!</div>
<div> 0 SNES Function norm 7.079201595974e-01</div>
<div> 1 SNES Function norm 1.247544399226e-02</div>
<div>24 TS dt 0.000150048 time 0.0382932</div>
<div>copy!</div>
<div> 0 SNES Function norm 6.197633272307e-01</div>
<div> 1 SNES Function norm 3.176914154297e-02</div>
<div>25 TS dt 0.000171939 time 0.0384432</div>
<div>copy!</div>
<div> 0 SNES Function norm 5.849423362557e-01</div>
<div> 1 SNES Function norm 9.462256964187e-03</div>
<div>26 TS dt 0.000176855 time 0.0386152</div>
<div>copy!</div>
<div> 0 SNES Function norm 4.304951653234e-01</div>
<div> 1 SNES Function norm 3.939112950208e-03</div>
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-- <br>
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<div>What most experimenters take for granted before they begin their experiments is infinitely more interesting than any results to which their experiments lead.<br>
-- Norbert Wiener</div>
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<div><a href="https://nam04.safelinks.protection.outlook.com/?url=http:%2F%2Fwww.cse.buffalo.edu%2F~knepley%2F&data=04%7C01%7Cskavou1%40lsu.edu%7C647e2e1d2354483d00cd08d903f72940%7C2d4dad3f50ae47d983a09ae2b1f466f8%7C0%7C0%7C637545182357863533%7CUnknown%7CTWFpbGZsb3d8eyJWIjoiMC4wLjAwMDAiLCJQIjoiV2luMzIiLCJBTiI6Ik1haWwiLCJXVCI6Mn0%3D%7C1000&sdata=U9kUKpcqDvvtZUvnvvydYXwABgzYihSzrApF3ik5dnQ%3D&reserved=0" originalsrc="http://www.cse.buffalo.edu/~knepley/" shash="XVbvVwHbtbO/7hBH4GEy/zB5gjmSKzD1VxAu+zayDg+5nMrVQCIIAiaRIn1ozuoP/5mnbhbumG41jGghm0Eyn3j6KcaQClSIyCBTC6CASRkSYYaAryhUCR3+cctatc0CBwx6DMJ52xPCfWPx5z3GskUZcaGjt4PVXOcbJ6PmZqE=" target="_blank">https://www.cse.buffalo.edu/~knepley/</a><br>
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