A Modified SSOR Preconditioning Strategy for Helmholtz Equations

<table class="table-group" id="tab3"><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>q</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">100</td><td align="center">120</td><td align="center">150</td><td align="center">160</td></tr><tr><td align="center" colspan="8"><hr/></td></tr><tr><td align="left" rowspan="4"><span style="width: 31.6px;"><svg height="14.85" id="M107" style="vertical-align:-2.22495pt" version="1.1" viewbox="0 0 31.6 14.85" width="31.6" 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="#x36"></use></g><g transform="matrix(.017,-0,0,-.017,23.385,12.012)"><use xlink:href="#x34"></use></g>
</svg></span></td><td align="center" rowspan="2"><span style="width: 39.012501px;"><svg height="15.4375" id="M108" 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"> 471</td><td align="center"> 456</td><td align="center">374</td><td align="center">349</td></tr><tr><td align="center">CPU(s)</td><td align="center"> 32.9344</td><td align="center">30.9688</td><td align="center"> 27.9844</td><td align="center">26.0156</td></tr><tr><td align="center" rowspan="2"><span style="width: 49.612499px;"><svg height="15.4375" id="M109" 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"> 373</td><td align="center"> 366</td><td align="center">347</td><td align="center">326</td></tr><tr><td align="center">CPU(s)</td><td align="center"> 12.4688</td><td align="center">12.2500</td><td align="center"> 11.6094</td><td align="center"> 10.9063</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="M103" style="vertical-align:-2.22495pt" version="1.1" viewbox="0 0 59.799999 15" width="59.799999" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
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q61 0 30 -130l-52 -220q-8 -33 4 -33q8 0 31 14.5t43 33.5z" id="x210E"></path></g><g transform="matrix(.017,-0,0,-.017,13.543,12.162)"><path d="M535 323h-483v50h483v-50zM535 138h-483v50h483v-50z" id="x3D"></path></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)"><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,-.017,51.569,12.162)"><use xlink:href="#x34"></use></g>
</svg> and <svg height="15.2875" id="M104" style="vertical-align:-3.50804pt" version="1.1" viewbox="0 0 62.262501 15.2875" width="62.262501" 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="M570 304q0 -108 -87 -199q-40 -42 -94.5 -74t-105.5 -43q-41 0 -65 11l-29 -141q-9 -45 -1.5 -58t45.5 -16l26 -2l-5 -29l-241 -10l4 26q51 10 67.5 24t26.5 60l113 520q-54 -20 -89 -41l-7 26q38 28 105 53l11 49q20 25 77 58l8 -7l-17 -77q39 14 102 14q82 0 119 -36
t37 -108zM482 289q0 114 -113 114q-26 0 -66 -7l-70 -327q12 -14 32 -25t39 -11q59 0 118.5 81.5t59.5 174.5z" id="x1D45D"></path></g><g transform="matrix(.017,-0,0,-.017,14.851,10.862)"><use xlink:href="#x3D"></use></g><g transform="matrix(.017,-0,0,-.017,29.555,10.862)"><use xlink:href="#x34"></use></g><g transform="matrix(.017,-0,0,-.017,37.715,10.862)"><use xlink:href="#x31"></use></g><g transform="matrix(.017,-0,0,-.017,45.874,10.862)"><path d="M241 635q53 0 94 -28.5t63.5 -76t33.5 -102.5t11 -116q0 -58 -11 -112.5t-34 -103.5t-63.5 -78.5t-94.5 -29.5t-95 28t-64.5 75t-34.5 102.5t-11 118.5q0 58 11.5 112.5t34.5 103t64.5 78t95.5 29.5zM238 602q-32 0 -55.5 -25t-35.5 -68t-17.5 -91t-5.5 -105
q0 -76 10 -138.5t37 -107.5t69 -45q32 0 55.5 25t35.5 68.5t17.5 91.5t5.5 105t-5.5 105.5t-18 92t-36 68t-56.5 24.5z" id="x30"></path></g><g transform="matrix(.017,-0,0,-.017,54.034,10.862)"><use xlink:href="#x30"></use></g>
</svg>, using COCG with the preconditioner <svg height="15.4375" id="M105" 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">
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</svg> and <svg height="15.4375" id="M106" 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">
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</svg>.

Journal of Applied Mathematics

A Modified SSOR Preconditioning Strategy for Helmholtz Equations

Table 3