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

Iteration counts for GMRES of <svg height="11.8375" id="M387" 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>, <svg height="20.0625" id="M388" style="vertical-align:-3.3907pt" version="1.1" viewbox="0 0 43.450001 20.0625" width="43.450001" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,15.775)"><path d="M856 669l7 -19q-33 -13 -66 -38q89 -68 89 -165q0 -67 -44 -122q-44 -53 -105 -53q-34 0 -53 18.5t-19 47.5q0 22 16 43.5t42 35.5l10 -18q-36 -21 -36 -54q0 -13 9 -21t25 -8q41 0 67.5 42.5t26.5 106.5q0 75 -55 123q-40 -38 -81 -119l-39 -81q-32 -70 -40 -86
q-83 -166 -179.5 -241.5t-219.5 -75.5q-77 0 -125 36t-48 87q0 31 17 51t42 20q22 0 36.5 -13.5t14.5 -34.5t-11.5 -34.5t-31.5 -13.5q-23 0 -30 12h-2q0 -28 35 -54q34 -25 85 -25q95 0 156 52q62 53 146 203l64 113q23 40 54 86q55 84 129 139q-76 48 -192 48
q-150 0 -250 -75q-101 -75 -101 -174q0 -41 30 -68.5t73 -27.5q158 0 205 249h20q-7 -112 -67 -196q-59 -83 -160 -83q-63 0 -102.5 35.5t-39.5 94.5q0 107 111 191t282 84q124 0 221 -59q38 24 84 41z" id="x1D4AB"></path></g>
<g transform="matrix(.012,-0,0,-.012,15.525,19.863)"><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,-.012,15.525,7.613)"><path d="M535 230h-483v50h483v-50z" id="x2212"></path></g><g transform="matrix(.012,-0,0,-.012,22.51,7.613)"><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,-.017,28.85,15.775)"><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>
</svg>, and <svg height="20.0625" id="M389" 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">
<g transform="matrix(.017,-0,0,-.017,.062,15.775)"><path d="M896 662l11 -15q-88 -73 -176 -73q-57 0 -181 30t-168 30q-93 0 -148 -39t-55 -104q0 -54 30 -83t77 -29q70 0 114 42.5t72 133.5l20 -1q-39 -205 -209 -205q-59 0 -102 37t-43 98q0 78 64 132q76 64 227 64q58 0 180 -28t174 -28q46 0 113 38zM657 568l14 -14
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>
</svg> for <svg height="12.4375" id="M390" style="vertical-align:-0.1638pt" version="1.1" viewbox="0 0 56.450001 12.4375" width="56.450001" 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)"><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(.017,-0,0,-.017,13.322,12.162)"><path d="M535 323h-483v50h483v-50zM535 138h-483v50h483v-50z" id="x3D"></path></g><g transform="matrix(.017,-0,0,-.017,28.025,12.162)"><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,-.017,36.185,12.162)"><path d="M113 -12q-24 0 -39.5 16t-15.5 42q0 24 16 40.5t40 16.5t40 -16.5t16 -40.5q0 -26 -16 -42t-41 -16z" id="x2E"></path></g><g transform="matrix(.017,-0,0,-.017,40.061,12.162)"><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,-.017,48.22,12.162)"><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>
</svg> and <svg height="12.4375" id="M391" style="vertical-align:-0.1638pt" version="1.1" viewbox="0 0 123.0375 12.4375" width="123.0375" 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)"><path d="M298 36l-68 -35q-25 -13 -42 -13q-98 0 -98 119v290h-69l-4 11l28 29h45v72l65 67l14 -2v-137h113q8 -8 6 -21t-12 -19h-107v-256q0 -96 59 -96q29 0 61 17z" id="x74"></path></g><g transform="matrix(.017,-0,0,-.017,5.145,12.162)"><path d="M257 449q92 0 154 -65t62 -158q0 -112 -67 -175t-150 -63q-98 0 -158.5 66.5t-60.5 154.5q0 59 21 106.5t54.5 75.5t71 43t73.5 15zM244 416q-48 0 -81 -47t-33 -128q0 -96 38 -158t99 -62q51 0 82 43.5t31 139.5q0 91 -36 151.5t-100 60.5z" id="x6F"></path></g><g transform="matrix(.017,-0,0,-.017,13.695,12.162)"><path d="M238 0h-223v26q48 5 60 17t12 61v473q0 50 -10.5 60.5t-58.5 15.5v24q73 8 148 35v-608q0 -49 11.5 -61t60.5 -17v-26z" id="x6C"></path></g><g transform="matrix(.017,-0,0,-.017,22.705,12.162)"><path d="M535 323h-483v50h483v-50zM535 138h-483v50h483v-50z" id="x3D"></path></g><g transform="matrix(.017,-0,0,-.017,37.409,12.162)"><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,-.017,45.568,12.162)"><path d="M113 -12q-24 0 -39.5 16t-15.5 42q0 24 16 40.5t40 16.5t40 -16.5t16 -40.5q0 -26 -16 -42t-41 -16z" id="x2E"></path></g><g transform="matrix(.017,-0,0,-.017,49.444,12.162)"><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,-.017,57.603,12.162)"><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,-.017,65.763,12.162)"><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,-.017,73.922,12.162)"><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,-.017,82.081,12.162)"><path d="M391 364q0 -28 -20.5 -53.5t-50 -43.5t-70 -34t-73.5 -25t-68 -17v-29q0 -48 20 -81t65 -33q73 0 157 76l16 -23q-107 -113 -221 -113q-21 0 -40.5 6.5t-39 22t-31.5 47t-12 75.5q0 69 33.5 139.5t95.5 114.5q77 55 143 55q42 0 69 -24.5t27 -59.5zM313 350
q0 26 -14.5 40.5t-37.5 14.5q-32 0 -44 -7q-82 -46 -104 -175q200 49 200 127z" id="x1D452"></path></g><g transform="matrix(.017,-0,0,-.017,92.893,12.162)"><path d="M535 230h-483v50h483v-50z" id="x2212"></path></g><g transform="matrix(.017,-0,0,-.017,106.645,12.162)"><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,-.017,114.804,12.162)"><path d="M456 178h-96v-72q0 -51 12.5 -62.5t72.5 -16.5v-27h-256v27q65 5 78 17t13 62v72h-260v28q182 271 300 426h40v-407h96v-47zM280 225v295h-2q-107 -148 -196 -295h198z" id="x34"></path></g>
</svg>.

International Scholarly Research Notices

tab3

Table 3

Table 3: New Preconditioning Techniques for Saddle Point Problems Arising from the Time-Harmonic Maxwell Equations 