A Modified SSOR Preconditioning Strategy for Helmholtz Equations

<table class="table-group" id="tab2"><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, 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">(1400, 40)</td><td align="center">(1500, 40)</td><td align="center">(1600, 40)</td><td align="center">(1700, 40)</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="M100" 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">
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</svg></span></td><td align="center"> IT</td><td align="center"> 230</td><td align="center"> 274</td><td align="center">310</td><td align="center">336</td></tr><tr><td align="center">CPU(s)</td><td align="center">3.2813</td><td align="center">3.9375</td><td align="center">4.8438</td><td align="center">4.7188</td></tr><tr><td align="center" rowspan="2"><span style="width: 49.612499px;"><svg height="15.4375" id="M102" 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></span></td><td align="center"> IT</td><td align="center"> 214</td><td align="center"> 228</td><td align="center"> 237</td><td align="center"> 249</td></tr><tr><td align="center">CPU(s)</td><td align="center"> 1.4375</td><td align="center">1.5156</td><td align="center"> 1.6563</td><td align="center"> 1.6250</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="M97" 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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</svg> meshes, using COCG with the preconditioner <svg height="15.4375" id="M98" 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="M99" 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> for solving the complex symmetric indefinite linear systems.

Journal of Applied Mathematics

A Modified SSOR Preconditioning Strategy for Helmholtz Equations

Table 2