A Modified SSOR Preconditioning Strategy for Helmholtz Equations

<table class="table-group" id="tab4"><tr><td><table class="table"><tr><td class="thead-hr" colspan="8"><hr/></td></tr><tr><td align="left"></td><td align="center" colspan="2"></td><td align="left"></td><td align="center" colspan="3"><i>p</i></td><td align="center"></td></tr><tr><td align="left"><i>h</i></td><td align="center" colspan="2">Preconditioner</td><td align="center">15000</td><td align="center">15500</td><td align="center">16000</td><td align="center">16500</td></tr><tr><td align="center" colspan="8"><hr/></td></tr><tr><td align="left" rowspan="4"><span style="width: 39.762501px;"><svg height="14.85" id="M114" style="vertical-align:-2.22495pt" version="1.1" viewbox="0 0 39.762501 14.85" width="39.762501" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,12.012)"><use xlink:href="#x31"></use></g><g transform="matrix(.017,-0,0,-.017,8.222,12.012)"><use xlink:href="#x2F"></use></g><g transform="matrix(.017,-0,0,-.017,15.225,12.012)"><use xlink:href="#x31"></use></g><g transform="matrix(.017,-0,0,-.017,23.385,12.012)"><use xlink:href="#x31"></use></g><g transform="matrix(.017,-0,0,-.017,31.544,12.012)"><use xlink:href="#x39"></use></g>
</svg></span></td><td align="center" rowspan="2"><span style="width: 39.012501px;"><svg height="15.4375" id="M115" style="vertical-align:-3.41931pt" version="1.1" viewbox="0 0 39.012501 15.4375" width="39.012501" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,11.112)"><use xlink:href="#x1D443"></use></g>
<g transform="matrix(.012,-0,0,-.012,10.875,15.187)"><use xlink:href="#x53"></use></g><g transform="matrix(.012,-0,0,-.012,16.515,15.187)"><use xlink:href="#x53"></use></g><g transform="matrix(.012,-0,0,-.012,22.155,15.187)"><use xlink:href="#x4F"></use></g><g transform="matrix(.012,-0,0,-.012,30.936,15.187)"><use xlink:href="#x52"></use></g>
</svg></span></td><td align="center"> IT</td><td align="center"> 148</td><td align="center"> 153</td><td align="center">147</td><td align="center">165</td></tr><tr><td align="center">CPU(s)</td><td align="center"> 55.3594</td><td align="center">57.4531</td><td align="center"> 57.8594</td><td align="center">52.4844</td></tr><tr><td align="center" rowspan="2"><span style="width: 49.612499px;"><svg height="15.4375" id="M116" style="vertical-align:-3.41931pt" version="1.1" viewbox="0 0 49.612499 15.4375" width="49.612499" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,11.112)"><use xlink:href="#x1D443"></use></g>
<g transform="matrix(.012,-0,0,-.012,10.875,15.187)"><use xlink:href="#x4D"></use></g><g transform="matrix(.012,-0,0,-.012,21.477,15.187)"><use xlink:href="#x53"></use></g><g transform="matrix(.012,-0,0,-.012,27.117,15.187)"><use xlink:href="#x53"></use></g><g transform="matrix(.012,-0,0,-.012,32.757,15.187)"><use xlink:href="#x4F"></use></g><g transform="matrix(.012,-0,0,-.012,41.538,15.187)"><use xlink:href="#x52"></use></g>
</svg></span></td><td align="center"> IT</td><td align="center"> 134</td><td align="center"> 137</td><td align="center"> 139</td><td align="center">144</td></tr><tr><td align="center">CPU(s)</td><td align="center"> 22.625</td><td align="center">22.9063</td><td align="center"> 23.2813</td><td align="center"> 23.7344</td></tr><tr class="table-tr"><td colspan="8"><hr class="tbody-hr"/></td></tr></table></td></tr></table>

Iteration numbers for the two-dimensional Helmholtz equations for <svg height="15" id="M110" style="vertical-align:-2.22495pt" version="1.1" viewbox="0 0 67.962502 15" width="67.962502" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,12.162)"><use xlink:href="#x210E"></use></g><g transform="matrix(.017,-0,0,-.017,13.543,12.162)"><use xlink:href="#x3D"></use></g><g transform="matrix(.017,-0,0,-.017,28.246,12.162)"><use xlink:href="#x31"></use></g><g transform="matrix(.017,-0,0,-.017,36.406,12.162)"><use xlink:href="#x2F"></use></g><g transform="matrix(.017,-0,0,-.017,43.409,12.162)"><use xlink:href="#x31"></use></g><g transform="matrix(.017,-0,0,-.017,51.569,12.162)"><use xlink:href="#x31"></use></g><g transform="matrix(.017,-0,0,-.017,59.728,12.162)"><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>
</svg> and <svg height="15.2875" id="M111" style="vertical-align:-3.50804pt" version="1.1" viewbox="0 0 60.525002 15.2875" width="60.525002" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,10.862)"><path d="M474 429q-19 -69 -47 -222l-65 -347q-8 -49 0 -60t49 -16l26 -3l-4 -26l-246 -12l4 25l17 3q37 6 50 20.5t23 60.5l67 321h-2q-65 -79 -150 -138q-69 -47 -104 -47q-28 0 -48.5 32t-20.5 81q0 78 35 148t90 117q67 57 161 77q23 5 58 5q43 0 90 -15zM387 387
q-30 16 -75 16q-58 0 -92 -27q-49 -39 -78.5 -112t-29.5 -136q0 -34 8.5 -52.5t21.5 -18.5q40 0 109.5 63.5t103.5 115.5q16 53 32 151z" id="x1D45E"></path></g><g transform="matrix(.017,-0,0,-.017,13.118,10.862)"><use xlink:href="#x3D"></use></g><g transform="matrix(.017,-0,0,-.017,27.821,10.862)"><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><g transform="matrix(.017,-0,0,-.017,35.981,10.862)"><use xlink:href="#x30"></use></g><g transform="matrix(.017,-0,0,-.017,44.14,10.862)"><use xlink:href="#x30"></use></g><g transform="matrix(.017,-0,0,-.017,52.3,10.862)"><use xlink:href="#x30"></use></g>
</svg>, using COCG with the preconditioner <svg height="15.4375" id="M112" style="vertical-align:-3.41931pt" version="1.1" viewbox="0 0 39.012501 15.4375" width="39.012501" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,11.112)"><use xlink:href="#x1D443"></use></g>
<g transform="matrix(.012,-0,0,-.012,10.875,15.187)"><use xlink:href="#x53"></use></g><g transform="matrix(.012,-0,0,-.012,16.515,15.187)"><use xlink:href="#x53"></use></g><g transform="matrix(.012,-0,0,-.012,22.155,15.187)"><use xlink:href="#x4F"></use></g><g transform="matrix(.012,-0,0,-.012,30.936,15.187)"><use xlink:href="#x52"></use></g>
</svg> and <svg height="15.4375" id="M113" style="vertical-align:-3.41931pt" version="1.1" viewbox="0 0 49.612499 15.4375" width="49.612499" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,11.112)"><use xlink:href="#x1D443"></use></g>
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</svg>. 

Journal of Applied Mathematics

A Modified SSOR Preconditioning Strategy for Helmholtz Equations

Table 4