New Preconditioning Techniques for Saddle Point Problems Arising from the Time-Harmonic Maxwell Equations

Comparison of iteration counts for GMRES of <svg height="11.8375" id="M375" style="vertical-align:-0.20474pt" version="1.1" viewbox="0 0 14.6625 11.8375" width="14.6625" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,11.525)"><path d="M837 672l9 -14q-102 -199 -265 -589q61 12 103 34l9 -16q-42 -32 -124 -45l-17 -42h-97l18 39q-91 6 -162 40q-91 -94 -173 -94q-48 0 -77.5 26t-29.5 66q0 32 17.5 52t44.5 20q21 0 35.5 -14t14.5 -34q0 -19 -12 -33.5t-31 -14.5q-22 0 -29 12h-2q0 -19 18.5 -34.5
t46.5 -15.5q41 0 74 17t76 60q-122 69 -122 207q0 83 57 132q56 49 130 49q103 0 161 -82l75 104q31 41 73 86q83 86 155 86q14 0 24 -2zM772 638h-1q-53 -19 -108 -81t-137 -186q32 -61 33 -144q129 283 213 411zM536 199q0 82 -29 145q-111 -171 -172 -239q69 -32 151 -38
l49 109q1 5 1 23zM492 372q-50 75 -143 75q-57 0 -102 -42q-44 -42 -44 -104q0 -117 107 -182q66 72 182 253z" id="x1D49C"></path></g>
</svg> and <svg height="20.0625" id="M376" style="vertical-align:-3.3907pt" version="1.1" viewbox="0 0 43.599998 20.0625" width="43.599998" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
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q-32 -32 -55 -80t-52 -137q69 14 121 15q17 26 39 41.5t39 15.5q23 0 23 -22q0 -15 -21 -37.5t-51 -37.5q-8 -24 -8 -60q0 -12 3 -26l-59 -20v18q0 31 11 70q-63 -3 -115 -12q-105 -282 -354 -282q-60 0 -104 28q-45 28 -45 83q0 32 16.5 50.5t43.5 18.5q20 0 35 -14.5
t15 -34.5q0 -21 -12 -35.5t-32 -14.5q-22 0 -29 13h-2q4 -27 34 -45t72 -18q76 0 132 48q56 47 121 177q-99 -32 -177 -85l-13 15q58 70 216 124l14 31q29 67 70 117.5t110 109.5z" id="x2131"></path></g>
<g transform="matrix(.012,-0,0,-.012,15.687,19.863)"><path d="M480 416q0 -21 -18 -41q-9 -11 -17 -7q-20 9 -42 9q-62 0 -140 -78q23 -69 88 -192q17 -31 27 -42t20 -11q16 0 62 46l17 -20q-64 -92 -119 -92q-35 0 -70 66q-41 73 -84 187q-36 -30 -62 -61q-27 -115 -35 -172q-41 -8 -78 -20l-6 6l140 612q7 28 0.5 34t-37.5 7l-34 1
l5 26q38 4 74 13.5t57 17t25 7.5q12 0 4 -32l-104 -443h2q35 38 97 93q39 35 65.5 56t62 41.5t58.5 20.5q19 0 30.5 -10t11.5 -22z" id="x1D458"></path></g><g transform="matrix(.012,-0,0,-.012,15.687,7.613)"><path d="M535 230h-483v50h483v-50z" id="x2212"></path></g><g transform="matrix(.012,-0,0,-.012,22.672,7.613)"><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,-.017,29,15.775)"><path d="M837 672l9 -14q-102 -199 -265 -589q61 12 103 34l9 -16q-42 -32 -124 -45l-17 -42h-97l18 39q-91 6 -162 40q-91 -94 -173 -94q-48 0 -77.5 26t-29.5 66q0 32 17.5 52t44.5 20q21 0 35.5 -14t14.5 -34q0 -19 -12 -33.5t-31 -14.5q-22 0 -29 12h-2q0 -19 18.5 -34.5
t46.5 -15.5q41 0 74 17t76 60q-122 69 -122 207q0 83 57 132q56 49 130 49q103 0 161 -82l75 104q31 41 73 86q83 86 155 86q14 0 24 -2zM772 638h-1q-53 -19 -108 -81t-137 -186q32 -61 33 -144q129 283 213 411zM536 199q0 82 -29 145q-111 -171 -172 -239q69 -32 151 -38
l49 109q1 5 1 23zM492 372q-50 75 -143 75q-57 0 -102 -42q-44 -42 -44 -104q0 -117 107 -182q66 72 182 253z" id="x1D49C"></path></g>
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<g transform="matrix(.017,-0,0,-.017,.062,12.162)"><path d="M480 416q0 -21 -18 -41q-9 -11 -17 -7q-20 9 -42 9q-62 0 -140 -78q23 -69 88 -192q17 -31 27 -42t20 -11q16 0 62 46l17 -20q-64 -92 -119 -92q-35 0 -70 66q-41 73 -84 187q-36 -30 -62 -61q-27 -115 -35 -172q-41 -8 -78 -20l-6 6l140 612q7 28 0.5 34t-37.5 7l-34 1
l5 26q38 4 74 13.5t57 17t25 7.5q12 0 4 -32l-104 -443h2q35 38 97 93q39 35 65.5 56t62 41.5t58.5 20.5q19 0 30.5 -10t11.5 -22z" id="x1D458"></path></g>
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International Scholarly Research Notices

tab2

Table 2

Table 2: New Preconditioning Techniques for Saddle Point Problems Arising from the Time-Harmonic Maxwell Equations 