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<p class="MsoNormal"><span style="font-size:11.0pt;mso-fareast-language:EN-US">I will solve the coupled problem in a non-linear fashion using SNES, so I will have one residual for each variable R(u1) and R(u2) and the resulting Jacobian matrix<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt;mso-fareast-language:EN-US"><o:p> </o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt;mso-fareast-language:EN-US">J = [J11 J12
<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt;mso-fareast-language:EN-US"> J21 J22] for each element.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt;mso-fareast-language:EN-US"><o:p> </o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt;mso-fareast-language:EN-US">For first-order Lagrange elements (in 2D) will have four nodes, then J would be of size 8 x 8. Let's say the variables are u and v. The order would be [u1 u2 u3 u4 v1 v2 v3 v4].<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt;mso-fareast-language:EN-US"><o:p> </o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt;mso-fareast-language:EN-US">Or should I rearrange the above entrées in the element Jacobian so as to have [u1 v1 u2 v2 u3 v3 u4 v4] ordering? In that case, the Jacobian will not have block ordering I guess.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt"><o:p> </o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt">In my case, the discretization of the u and v are from the same element space.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt">Can you please elaborate when you say, “Just enlarge the block size of your matrix”?
<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt">I have never used block size before and not sure what it accomplishes.
<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt;mso-fareast-language:EN-US">Do you mean to say, I should use MATBAIJ instead of MATAIJ?
<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt;mso-fareast-language:EN-US"><o:p> </o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt;mso-fareast-language:EN-US">Kind regards,<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt;mso-fareast-language:EN-US">Karthik.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt;mso-fareast-language:EN-US"><o:p> </o:p></span></p>
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<p class="MsoNormal" style="margin-bottom:12.0pt"><b><span style="font-size:12.0pt;color:black">From:
</span></b><span style="font-size:12.0pt;color:black">Matthew Knepley <knepley@gmail.com><br>
<b>Date: </b>Sunday, 10 September 2023 at 19:03<br>
<b>To: </b>Chockalingam, Karthikeyan (STFC,DL,HC) <karthikeyan.chockalingam@stfc.ac.uk><br>
<b>Cc: </b>petsc-users@mcs.anl.gov <petsc-users@mcs.anl.gov><br>
<b>Subject: </b>Re: [petsc-users] Coupled system of equations on unstructured mesh<o:p></o:p></span></p>
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<p class="MsoNormal"><span style="font-size:11.0pt">On Sun, Sep 10, 2023 at 1:48 PM Karthikeyan Chockalingam - STFC UKRI via petsc-users <<a href="mailto:petsc-users@mcs.anl.gov">petsc-users@mcs.anl.gov</a>> wrote:<o:p></o:p></span></p>
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<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span style="font-size:11.0pt">Hello,<o:p></o:p></span></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span style="font-size:11.0pt"> <o:p></o:p></span></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span style="font-size:11.0pt">I have so far solved scalar field problems using finite elements on a given (<b>unstructured</b>) mesh. I made use of
<span style="color:black">MATMPIAIJ</span> to create matrixes, <span class="m8670309824937278731s1">
MatCreateVecs</span>(<span class="m8670309824937278731s2">A</span>, &<span class="m8670309824937278731s2">b</span>, &<span class="m8670309824937278731s2">x</span>) to create vectors, and
<span style="color:#6C36A9">MatZeroRowsColumnsIS </span>to set boundary conditions.<o:p></o:p></span></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span style="font-size:11.0pt"> <o:p></o:p></span></p>
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<span style="font-size:11.0pt;color:black">Now, I would like to solve a coupled system of equations for the quantities u1 and u2 on the (<b>unstructured</b>) mesh. I.e., the matrix should get the double number of rows and columns,
</span><span style="font-size:11.0pt"><o:p></o:p></span></p>
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<span style="font-size:11.0pt;color:black"> </span><span style="font-size:11.0pt"><o:p></o:p></span></p>
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<span style="font-size:11.0pt;color:black">A = [A00 A01 </span><span style="font-size:11.0pt"><o:p></o:p></span></p>
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<span style="font-size:11.0pt;color:black"> A10 A11]</span><span style="font-size:11.0pt"><o:p></o:p></span></p>
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<p class="MsoNormal"><span style="font-size:11.0pt"><o:p> </o:p></span></p>
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<p class="MsoNormal"><span style="font-size:11.0pt">This is usually not a good way to think of it. This division means that all variables of one field come before<o:p></o:p></span></p>
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<p class="MsoNormal"><span style="font-size:11.0pt">all those of another. It is much more common to group together all the unknowns at a given point.<o:p></o:p></span></p>
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<p class="MsoNormal"><span style="font-size:11.0pt"><o:p> </o:p></span></p>
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<p class="MsoNormal"><span style="font-size:11.0pt">You could, if the discretizations of u1 and u2 are the same, just enlarge the blocksize of your matrix. Then you set u1 and u2 for each vec location, or a 2x2 block for each Jacobian location.<o:p></o:p></span></p>
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<p class="MsoNormal"><span style="font-size:11.0pt"><o:p> </o:p></span></p>
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<p class="MsoNormal"><span style="font-size:11.0pt"> Thanks,<o:p></o:p></span></p>
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<p class="MsoNormal"><span style="font-size:11.0pt"><o:p> </o:p></span></p>
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<p class="MsoNormal"><span style="font-size:11.0pt"> Matt<o:p></o:p></span></p>
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<p class="MsoNormal"><span style="font-size:11.0pt"> <o:p></o:p></span></p>
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<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;background:white">
<span style="font-size:11.0pt;color:black">the vectors contain twice the number of entries (e.g. first all u1s and then all u2s). I would like to be sure that the entries of u1 and u2, which are associated with the same element are located on the same processor.</span><span style="font-size:11.0pt"><o:p></o:p></span></p>
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<span style="font-size:11.0pt;color:black"> </span><span style="font-size:11.0pt"><o:p></o:p></span></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;background:white">
<span style="font-size:11.0pt;color:black">Is a pre-defined structure already available within PETSc to enlarge such a single equation to store the entries of coupled equations?</span><span style="font-size:11.0pt"><o:p></o:p></span></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span style="font-size:11.0pt"> <o:p></o:p></span></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span style="font-size:11.0pt">-\Delta u_1+c_{11} u_1+c_{12} u_2=f_1<o:p></o:p></span></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span style="font-size:11.0pt">-\Delta u_2+c_{21} u_1+c_{22} u_2=f_2<o:p></o:p></span></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span style="font-size:11.0pt"> <o:p></o:p></span></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;background:white">
<span style="font-size:11.0pt;color:black">Would I still be able to use </span><span style="font-size:11.0pt;color:#6C36A9">MatZeroRowsColumnsIS
</span><span style="font-size:11.0pt;color:black">u1 and u2 independently to enforce boundary conditions?
</span><span style="font-size:11.0pt;color:#6C36A9">MatZeroRowsColumnsIS</span><span style="font-size:11.0pt;color:black">(</span><span style="font-size:11.0pt;color:#326D74">A</span><span style="font-size:11.0pt;color:black">,
</span><span style="font-size:11.0pt;color:#326D74">is</span><span style="font-size:11.0pt;color:black">,
</span><span style="font-size:11.0pt;color:#1C00CF">1</span><span style="font-size:11.0pt;color:black">,
</span><span style="font-size:11.0pt;color:#326D74">x</span><span style="font-size:11.0pt;color:black">,
</span><span style="font-size:11.0pt;color:#326D74">b</span><span style="font-size:11.0pt;color:black">);</span><span style="font-size:11.0pt"><o:p></o:p></span></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span style="font-size:11.0pt"> <o:p></o:p></span></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span style="font-size:11.0pt">I don’t know where to begin. I have so far only been exposed to using
<span style="color:black">MATMPIAIJ and MatSetValues to create and assign values to matrix entries respectively.
</span>I would be grateful if you could provide the stepwise guide.<o:p></o:p></span></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span style="font-size:11.0pt"> <o:p></o:p></span></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span style="font-size:11.0pt">Kind regards,<o:p></o:p></span></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span style="font-size:11.0pt">Karthik.<o:p></o:p></span></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span style="font-size:11.0pt"> <o:p></o:p></span></p>
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<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span style="font-size:11.0pt">-- <o:p></o:p></span></p>
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<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><b><span style="font-size:10.5pt;font-family:"Arial",sans-serif;color:#2D2D62">Karthik Chockalingam, Ph.D.</span></b><span style="font-size:11.0pt"><o:p></o:p></span></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span style="font-size:10.5pt;font-family:"Arial",sans-serif;color:black">Senior Research Software Engineer</span><span style="font-size:11.0pt"><o:p></o:p></span></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span style="font-size:10.5pt;font-family:"Arial",sans-serif;color:black">High Performance Systems Engineering Group</span><span style="font-size:11.0pt"><o:p></o:p></span></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span style="font-size:10.5pt;font-family:"Arial",sans-serif;color:black">Hartree Centre | Science and Technology Facilities Council</span><span style="font-size:11.0pt"><o:p></o:p></span></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span style="font-size:10.5pt;font-family:"Arial",sans-serif;color:#212121"><a href="mailto:karthikeyan.chockalingam@stfc.ac.uk" target="_blank">karthikeyan.chockalingam@stfc.ac.uk</a></span><span style="font-size:11.0pt"><o:p></o:p></span></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span style="font-size:12.0pt;font-family:"Times New Roman",serif;color:black"> </span><span style="font-size:10.5pt;font-family:"Arial",sans-serif;color:#414141"> </span><span style="font-size:11.0pt"><o:p></o:p></span></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><b><span style="font-size:10.5pt;font-family:"Arial",sans-serif;color:#414141"> </span></b><span style="font-size:12.0pt;font-family:"Times New Roman",serif;color:#1F497D"><b>Error!
Filename not specified.</b></span><span style="font-size:11.0pt"><o:p></o:p></span></p>
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<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span style="font-size:11.0pt"> <o:p></o:p></span></p>
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<o:p></o:p></span></p>
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<p class="MsoNormal"><span style="font-size:11.0pt"><o:p> </o:p></span></p>
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<p class="MsoNormal"><span class="gmailsignatureprefix"><span style="font-size:11.0pt">--
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<p class="MsoNormal"><span style="font-size:11.0pt">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<o:p></o:p></span></p>
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<p class="MsoNormal"><span style="font-size:11.0pt"><o:p> </o:p></span></p>
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<p class="MsoNormal"><span style="font-size:11.0pt"><a href="http://www.cse.buffalo.edu/~knepley/" target="_blank">https://www.cse.buffalo.edu/~knepley/</a><o:p></o:p></span></p>
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