A Parameterized Splitting Preconditioner for Generalized Saddle Point Problems

<table class="table-group" id="tab6"><tr><td><table class="table"><tr><td class="thead-hr" colspan="6"><hr/></td></tr><tr class="thead"><td align="left">  Grid </td><td align="center"><span style="width: 9.7749996px;"><svg height="7.9499998" id="M124" style="vertical-align:-0.1638pt" version="1.1" viewbox="0 0 9.7749996 7.9499998" width="9.7749996" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
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</svg></span></td><td align="center"> its </td><td align="center"> LU time </td><td align="center"> its time </td><td align="center"> Total time </td></tr><tr><td class="thead-hr" colspan="6"><hr/></td></tr><tr><td align="left"><span style="width: 50.299999px;"><svg height="11.225" id="M125" style="vertical-align:-0.1638pt" version="1.1" viewbox="0 0 50.299999 11.225" width="50.299999" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,10.963)"><path d="M384 0h-275v27q67 5 81.5 18.5t14.5 68.5v385q0 38 -7.5 47.5t-40.5 10.5l-48 2v24q85 15 178 52v-521q0 -55 14.5 -68.5t82.5 -18.5v-27z" id="x31"></path></g><g transform="matrix(.017,-0,0,-.017,8.222,10.963)"><path d="M137 343l67 33q37 17 63 17q79 0 129.5 -53t50.5 -131q0 -92 -58 -156.5t-147 -64.5t-147 68t-58 182q0 63 17 119t43 95.5t61.5 72t69 52t67.5 31.5q62 22 128 33l6 -32q-56 -11 -108 -35q-149 -71 -184 -231zM227 337q-47 0 -95 -27q-6 -23 -6 -70q0 -93 36 -155.5
t96 -62.5q53 0 78 45.5t25 105.5q0 68 -35 116t-99 48z" id="x36"></path></g><g transform="matrix(.017,-0,0,-.017,20.155,10.963)"><path d="M528 54l-36 -38l-198 201l-198 -201l-36 38l197 200l-197 201l36 38l198 -202l198 202l36 -38l-197 -201z" id="xD7"></path></g><g transform="matrix(.017,-0,0,-.017,33.907,10.963)"><path d="M384 0h-275v27q67 5 81.5 18.5t14.5 68.5v385q0 38 -7.5 47.5t-40.5 10.5l-48 2v24q85 15 178 52v-521q0 -55 14.5 -68.5t82.5 -18.5v-27z" id="x31"></path></g><g transform="matrix(.017,-0,0,-.017,42.066,10.963)"><path d="M137 343l67 33q37 17 63 17q79 0 129.5 -53t50.5 -131q0 -92 -58 -156.5t-147 -64.5t-147 68t-58 182q0 63 17 119t43 95.5t61.5 72t69 52t67.5 31.5q62 22 128 33l6 -32q-56 -11 -108 -35q-149 -71 -184 -231zM227 337q-47 0 -95 -27q-6 -23 -6 -70q0 -93 36 -155.5
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</svg></span></td><td align="center"> 10000</td><td align="center"> 4 </td><td align="center">0.0458</td><td align="center"> 0.0223</td><td align="center"> 0.0681 </td></tr><tr><td align="left"><span style="width: 50.299999px;"><svg height="11.125" id="M126" style="vertical-align:-0.1638pt" version="1.1" viewbox="0 0 50.299999 11.125" width="50.299999" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
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q0 45 -26.5 69.5t-68.5 24.5q-67 0 -120 -79l-20 20l43 63q51 56 127 56h1q66 0 107 -37t41 -95q0 -42 -31 -71q-22 -23 -68 -54z" id="x33"></path></g><g transform="matrix(.017,-0,0,-.017,42.066,10.862)"><path d="M412 140l28 -9q0 -2 -35 -131h-373v23q112 112 161 170q59 70 92 127t33 115q0 63 -31 98t-86 35q-75 0 -137 -93l-22 20l57 81q55 59 135 59q69 0 118.5 -46.5t49.5 -122.5q0 -62 -29.5 -114t-102.5 -130l-141 -149h186q42 0 58.5 10.5t38.5 56.5z" id="x32"></path></g>
</svg></span></td><td align="center"> 10000</td><td align="center"> 4 </td><td align="center">0.5748</td><td align="center">0.0670</td><td align="center"> 0.6418</td></tr><tr><td align="left"><span style="width: 50.299999px;"><svg height="11.225" id="M127" style="vertical-align:-0.1638pt" version="1.1" viewbox="0 0 50.299999 11.225" width="50.299999" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,10.963)"><path d="M137 343l67 33q37 17 63 17q79 0 129.5 -53t50.5 -131q0 -92 -58 -156.5t-147 -64.5t-147 68t-58 182q0 63 17 119t43 95.5t61.5 72t69 52t67.5 31.5q62 22 128 33l6 -32q-56 -11 -108 -35q-149 -71 -184 -231zM227 337q-47 0 -95 -27q-6 -23 -6 -70q0 -93 36 -155.5
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</svg></span></td><td align="center"> 10000</td><td align="center"> 5</td><td align="center">12.2642 </td><td align="center"> 0.4179</td><td align="center"> 12.6821</td></tr><tr><td align="left"><span style="width: 66.612503px;"><svg height="11.125" id="M128" style="vertical-align:-0.1638pt" version="1.1" viewbox="0 0 66.612503 11.125" width="66.612503" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
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</svg></span></td><td align="center"> 10000</td><td align="center"> 7 </td><td align="center">128.2758</td><td align="center"> 1.6275</td><td align="center">129.9033</td></tr><tr class="table-tr"><td colspan="6"><hr class="tbody-hr"/></td></tr></table></td></tr></table>

Preconditioned GMRES(20) on steady Oseen problems with different grid sizes (uniform grids), viscosity <svg height="11.125" id="M123" style="vertical-align:-0.1638pt" version="1.1" viewbox="0 0 47.900002 11.125" width="47.900002" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
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</svg>.

Journal of Applied Mathematics

tab6

Table 6

Table 6: A Parameterized Splitting Preconditioner for Generalized Saddle Point Problems 