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<p style="margin-top:0;margin-bottom:0"><span>Dear Sir or Madam,</span><br>
</p>
<p style="margin-top:0;margin-bottom:0"><span><br>
</span></p>
<p style="margin-top:0;margin-bottom:0"><span>Recently, our research group try to use the
<span>PCfieldsplit method in Petsc to construct the preconditioner for the linear equation system. For example, if we have following system,</span></span></p>
<p style="margin-top:0;margin-bottom:0"><span><span></p>
<p style="font-family: Calibri, Helvetica, sans-serif, EmojiFont, "Apple Color Emoji", "Segoe UI Emoji", NotoColorEmoji, "Segoe UI Symbol", "Android Emoji", EmojiSymbols; font-size: 16px;">
<span style="font-size: 12pt;">K11, K12, K13 u1 f1</span></p>
<p style="font-family: Calibri, Helvetica, sans-serif, EmojiFont, "Apple Color Emoji", "Segoe UI Emoji", NotoColorEmoji, "Segoe UI Symbol", "Android Emoji", EmojiSymbols; font-size: 16px;">
<span style="font-size: 12pt;"> K21, K22, K23 u2 = f2 (1)</span></p>
<p style="font-family: Calibri, Helvetica, sans-serif, EmojiFont, "Apple Color Emoji", "Segoe UI Emoji", NotoColorEmoji, "Segoe UI Symbol", "Android Emoji", EmojiSymbols; font-size: 16px;">
<span style="font-size: 12pt;"> K31, K32, K33 u3 f3</span></p>
<p style="font-family: Calibri, Helvetica, sans-serif, EmojiFont, "Apple Color Emoji", "Segoe UI Emoji", NotoColorEmoji, "Segoe UI Symbol", "Android Emoji", EmojiSymbols; font-size: 16px;">
<span style="font-size: 12pt;"> </span></p>
<p style="font-family: Calibri, Helvetica, sans-serif, EmojiFont, "Apple Color Emoji", "Segoe UI Emoji", NotoColorEmoji, "Segoe UI Symbol", "Android Emoji", EmojiSymbols; font-size: 16px;">
<span style="font-size: 12pt;">we construct the preconditioner as follows:</span></p>
<p style="font-family: Calibri, Helvetica, sans-serif, EmojiFont, "Apple Color Emoji", "Segoe UI Emoji", NotoColorEmoji, "Segoe UI Symbol", "Android Emoji", EmojiSymbols; font-size: 16px;">
<span style="font-size: 12pt;"><span style="color: rgb(33, 33, 33); font-size: 14.67px; font-family: wf_segoe-ui_normal, "Segoe UI", "Segoe WP", Tahoma, Arial, sans-serif, serif, EmojiFont;"></span></span></p>
<p style="font-family: Calibri, Helvetica, sans-serif, EmojiFont, "Apple Color Emoji", "Segoe UI Emoji", NotoColorEmoji, "Segoe UI Symbol", "Android Emoji", EmojiSymbols; font-size: 16px;">
<span style="font-size: 12pt;"> K11, K12, K13 </span></p>
<p style="font-family: Calibri, Helvetica, sans-serif, EmojiFont, "Apple Color Emoji", "Segoe UI Emoji", NotoColorEmoji, "Segoe UI Symbol", "Android Emoji", EmojiSymbols; font-size: 16px;">
<span style="font-size: 12pt;"> 0, K22, K23 </span></p>
<p style="font-family: Calibri, Helvetica, sans-serif, EmojiFont, "Apple Color Emoji", "Segoe UI Emoji", NotoColorEmoji, "Segoe UI Symbol", "Android Emoji", EmojiSymbols; font-size: 16px;">
<span style="font-size: 12pt;"> 0, 0, K33. </span></p>
<p style="font-family: Calibri, Helvetica, sans-serif, EmojiFont, "Apple Color Emoji", "Segoe UI Emoji", NotoColorEmoji, "Segoe UI Symbol", "Android Emoji", EmojiSymbols; font-size: 16px;">
<span style="font-size: 12pt;"><br>
</span></p>
<p style="font-family: Calibri, Helvetica, sans-serif, EmojiFont, "Apple Color Emoji", "Segoe UI Emoji", NotoColorEmoji, "Segoe UI Symbol", "Android Emoji", EmojiSymbols; font-size: 16px;">
We also can reorder the system (1):</p>
<p style="font-family: Calibri, Helvetica, sans-serif, EmojiFont, "Apple Color Emoji", "Segoe UI Emoji", NotoColorEmoji, "Segoe UI Symbol", "Android Emoji", EmojiSymbols; font-size: 16px;">
<span style="color: rgb(33, 33, 33); font-family: wf_segoe-ui_normal, "Segoe UI", "Segoe WP", Tahoma, Arial, sans-serif, serif, EmojiFont; font-size: 14.6667px;"> K22, K21, K23 u2 f2</span><br style="color: rgb(33, 33, 33); font-family: wf_segoe-ui_normal, "Segoe UI", "Segoe WP", Tahoma, Arial, sans-serif, serif, EmojiFont; font-size: 14.6667px;">
<span style="color: rgb(33, 33, 33); font-family: wf_segoe-ui_normal, "Segoe UI", "Segoe WP", Tahoma, Arial, sans-serif, serif, EmojiFont; font-size: 14.6667px;"> K12, K11, K13 u1 = f1 (2)</span><br style="color: rgb(33, 33, 33); font-family: wf_segoe-ui_normal, "Segoe UI", "Segoe WP", Tahoma, Arial, sans-serif, serif, EmojiFont; font-size: 14.6667px;">
<span style="color: rgb(33, 33, 33); font-family: wf_segoe-ui_normal, "Segoe UI", "Segoe WP", Tahoma, Arial, sans-serif, serif, EmojiFont; font-size: 14.6667px;"> K32, K31, K33 u3 f3</span><br>
</p>
<p style="font-family: Calibri, Helvetica, sans-serif, EmojiFont, "Apple Color Emoji", "Segoe UI Emoji", NotoColorEmoji, "Segoe UI Symbol", "Android Emoji", EmojiSymbols; font-size: 16px;">
<span style="color: rgb(33, 33, 33); font-family: wf_segoe-ui_normal, "Segoe UI", "Segoe WP", Tahoma, Arial, sans-serif, serif, EmojiFont; font-size: 14.6667px;"><br>
</span></p>
<p style="font-family: Calibri, Helvetica, sans-serif, EmojiFont, "Apple Color Emoji", "Segoe UI Emoji", NotoColorEmoji, "Segoe UI Symbol", "Android Emoji", EmojiSymbols; font-size: 16px;">
<span style="color: rgb(33, 33, 33); font-family: wf_segoe-ui_normal, "Segoe UI", "Segoe WP", Tahoma, Arial, sans-serif, serif, EmojiFont; font-size: 14.6667px;">with the preconditioner </span></p>
<p style="font-family: Calibri, Helvetica, sans-serif, EmojiFont, "Apple Color Emoji", "Segoe UI Emoji", NotoColorEmoji, "Segoe UI Symbol", "Android Emoji", EmojiSymbols; font-size: 16px;">
<span style="color: rgb(33, 33, 33); font-family: wf_segoe-ui_normal, "Segoe UI", "Segoe WP", Tahoma, Arial, sans-serif, serif, EmojiFont; font-size: 14.6667px;"><span style="color: rgb(33, 33, 33); font-family: wf_segoe-ui_normal, "Segoe UI", "Segoe WP", Tahoma, Arial, sans-serif, serif, EmojiFont; font-size: 14.6667px;">K22,
K21, K23 </span><br style="color: rgb(33, 33, 33); font-family: wf_segoe-ui_normal, "Segoe UI", "Segoe WP", Tahoma, Arial, sans-serif, serif, EmojiFont; font-size: 14.6667px;">
<span style="color: rgb(33, 33, 33); font-family: wf_segoe-ui_normal, "Segoe UI", "Segoe WP", Tahoma, Arial, sans-serif, serif, EmojiFont; font-size: 14.6667px;"> 0, K11, K13 </span><br style="color: rgb(33, 33, 33); font-family: wf_segoe-ui_normal, "Segoe UI", "Segoe WP", Tahoma, Arial, sans-serif, serif, EmojiFont; font-size: 14.6667px;">
<span style="color: rgb(33, 33, 33); font-family: wf_segoe-ui_normal, "Segoe UI", "Segoe WP", Tahoma, Arial, sans-serif, serif, EmojiFont; font-size: 14.6667px;"> 0, 0, K33</span><br>
</span></p>
<p style="font-family: Calibri, Helvetica, sans-serif, EmojiFont, "Apple Color Emoji", "Segoe UI Emoji", NotoColorEmoji, "Segoe UI Symbol", "Android Emoji", EmojiSymbols; font-size: 16px;">
<span style="color: rgb(33, 33, 33); font-family: wf_segoe-ui_normal, "Segoe UI", "Segoe WP", Tahoma, Arial, sans-serif, serif, EmojiFont; font-size: 14.6667px;"><span style="color: rgb(33, 33, 33); font-family: wf_segoe-ui_normal, "Segoe UI", "Segoe WP", Tahoma, Arial, sans-serif, serif, EmojiFont; font-size: 14.6667px;"><br>
</span></span></p>
<p style="font-family: Calibri, Helvetica, sans-serif, EmojiFont, "Apple Color Emoji", "Segoe UI Emoji", NotoColorEmoji, "Segoe UI Symbol", "Android Emoji", EmojiSymbols; font-size: 16px;">
<span style="color: rgb(33, 33, 33); font-family: wf_segoe-ui_normal, "Segoe UI", "Segoe WP", Tahoma, Arial, sans-serif, serif, EmojiFont; font-size: 14.6667px;"><span style="color: rgb(33, 33, 33); font-family: wf_segoe-ui_normal, "Segoe UI", "Segoe WP", Tahoma, Arial, sans-serif, serif, EmojiFont; font-size: 14.6667px;">We
use the same solver for Eq.(1) and Eq. (2) with corresponding preconditioner. The question is: is solving the Eq. (1)
<span style="color: rgb(33, 33, 33); font-family: wf_segoe-ui_normal, "Segoe UI", "Segoe WP", Tahoma, Arial, sans-serif, serif, EmojiFont; font-size: 14.6667px;">
equivalent to</span></span></span><span style="font-size: 12pt;"> solving Eq. (2) in Petsc? From our results, it seems not. Thank you so much.</span></p>
<p style="font-family: Calibri, Helvetica, sans-serif, EmojiFont, "Apple Color Emoji", "Segoe UI Emoji", NotoColorEmoji, "Segoe UI Symbol", "Android Emoji", EmojiSymbols; font-size: 16px;">
<span style="font-size: 12pt;"><br>
</span></p>
<p style="font-family: Calibri, Helvetica, sans-serif, EmojiFont, "Apple Color Emoji", "Segoe UI Emoji", NotoColorEmoji, "Segoe UI Symbol", "Android Emoji", EmojiSymbols; font-size: 16px;">
<span style="font-size: 12pt;">Best,</span></p>
<p style="font-family: Calibri, Helvetica, sans-serif, EmojiFont, "Apple Color Emoji", "Segoe UI Emoji", NotoColorEmoji, "Segoe UI Symbol", "Android Emoji", EmojiSymbols; font-size: 16px;">
<span style="font-size: 12pt;">Guoyi </span></p>
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</span></span></p>
<p style="margin-top:0;margin-bottom:0"><span><span><br>
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