Incomplete Augmented Lagrangian Preconditioner for Steady Incompressible Navier-Stokes Equations

<table class="table-group" id="tab10"><tr><td><table class="table"><tr><td class="thead-hr" colspan="4"><hr/></td></tr><tr class="thead"><td align="left" rowspan="2">   Grid </td><td align="center" colspan="3">Viscosity</td></tr><tr class="thead"><td align="center"> 0.1 </td><td align="center"> 0.01 </td><td align="center"> 0.005 </td></tr><tr><td class="thead-hr" colspan="4"><hr/></td></tr><tr><td align="left"><span style="width: 50.299999px;"><svg height="11.225" id="M164" style="vertical-align:-0.3135pt" 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">
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</svg></span></td><td align="center">12 (0.5)</td><td align="center">15 (0.2)</td><td align="center">26 (0.1)</td></tr><tr><td align="left"><span style="width: 50.299999px;"><svg height="11.275" id="M165" style="vertical-align:-0.3135pt" version="1.1" viewbox="0 0 50.299999 11.275" 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,-.017,8.222,10.962)"><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><g transform="matrix(.017,-0,0,-.017,20.155,10.962)"><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,-.017,33.907,10.962)"><path d="M244 635q90 0 143 -72t53 -177q0 -133 -65 -229.5t-171 -139.5q-79 -32 -140 -32l-5 30q109 18 185 91t101 186l-68 -36q-29 -16 -60 -16q-79 0 -129 51.5t-50 130.5q0 80 57 146.5t149 66.5zM228 602q-52 0 -78 -45.5t-26 -98.5q0 -69 36.5 -115.5t97.5 -46.5
q53 0 90 28q4 31 4 66q0 51 -9.5 95.5t-39 80.5t-75.5 36z" id="x39"></path></g><g transform="matrix(.017,-0,0,-.017,42.066,10.962)"><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">13 (0.5)</td><td align="center">16 (0.08)</td><td align="center">25 (0.1)</td></tr><tr><td align="left"><span style="width: 58.450001px;"><svg height="11.225" id="M166" style="vertical-align:-0.3135pt" version="1.1" viewbox="0 0 58.450001 11.225" width="58.450001" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
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t98.5 -62.5q61 48 61 124q0 59 -29.5 90t-69.5 31zM248 20q46 0 76.5 33.5t30.5 89.5q0 50 -39 85.5t-110 71.5q-81 -54 -81 -137q0 -67 35.5 -105t87.5 -38z" id="x38"></path></g><g transform="matrix(.017,-0,0,-.017,50.226,10.963)"><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">12 (0.5)</td><td align="center">15 (0.08)</td><td align="center">27 (0.1)</td></tr><tr class="table-tr"><td colspan="4"><hr class="tbody-hr"/></td></tr></table></td></tr></table>

GMRES iterations with incomplete AL preconditioner for the steady Oseen in a uniform backward facing step  (Q2-Q1 FEM and Newton).

The Scientific World Journal

tab10

Table 10

Table 10: Incomplete Augmented Lagrangian Preconditioner for Steady Incompressible Navier-Stokes Equations 