A Numerical Method for Solving Elliptic Interface Problems Using Petrov-Galerkin Formulation with Adaptive Refinement

<table class="fixed-width table-group" id="tab1"><tr><td><table class="table"><colgroup><col style="width:11.07em"/><col style="width:4.05em"/><col style="width:4.03em"/><col style="width:5.59em"/></colgroup><tr><td class="thead-hr" colspan="4"><hr/></td></tr><tr class="thead"><td class="align_left">Mesh </td><td class="align_center"><span style="width: 11.0475ptpx;"><svg height="8.8423pt" id="M170" style="vertical-align:-0.2064009pt" version="1.1" viewbox="-0.0498162 -8.6359 11.0475 8.8423" width="11.0475pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M822 650H589L583 622C660 617 677 607 674 561C672 534 664 481 647 390L600 137H596L273 650H126L120 622C176 620 194 615 207 594C221 571 225 557 214 504L161 257C141 166 129 112 121 85C108 42 83 30 29 28L23 0H260L266 28C193 33 173 42 176 89C178 122 186 172 202 255L256 527H259L583 -8H612L690 390C708 481 720 535 728 558C744 603 756 619 816 622L822 650Z" id="g113-79"></path></g></svg></span></td><td class="align_center"><span style="width: 31.6821ptpx;"><svg height="15.0991pt" id="M171" style="vertical-align:-4.519pt" version="1.1" viewbox="-0.0498162 -10.5801 31.6821 15.0991" width="31.6821pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,-6.668)"><path d="M162 0V316H102V0H162ZM309 0V316H249V0H309Z" id="g119-35"></path></g><g transform="matrix(.013,0,0,-0.013,0,-3.066)"><path d="M162 0V316H102V0H162ZM309 0V316H249V0H309Z" id="g119-35"></path></g><g transform="matrix(.013,0,0,-0.013,0,.535)"><path d="M162 0V316H102V0H162ZM309 0V316H249V0H309Z" id="g119-35"></path></g><g transform="matrix(.013,0,0,-0.013,0,4.136)"><path d="M162 0V316H102V0H162ZM309 0V316H249V0H309Z" id="g119-35"></path></g><g transform="matrix(.013,0,0,-0.013,5.363,0)"><path d="M391 364C391 409 353 448 295 448C249 448 198 426 152 393C65 331 23 225 23 139C23 14 96 -12 146 -12C198 -12 280 9 367 101L351 124C300 78 242 48 194 48C129 48 109 107 109 162V191C208 213 391 266 391 364ZM313 350C313 305 268 261 113 223C132 334 187 381 217 398C227 404 244 405 261 405C290 405 313 385 313 350Z" id="g113-102"></path></g><g transform="matrix(.0091,0,0,-0.0091,10.68,3.132)"><path d="M528 97L513 125C484 95 440 68 431 68C425 68 421 76 427 107C449 227 476 342 505 451H495L418 424L385 278C338 191 220 62 157 62C145 62 133 76 145 130L200 368C215 434 210 451 183 451C157 451 92 418 24 352L37 324C78 356 106 377 115 377C123 377 122 364 116 340L64 115C58 89 56 68 56 52C56 2 78 -12 103 -12C127 -12 159 -4 190 18C247 59 309 113 366 187H368L351 108C328 1 348 -12 368 -12C403 -12 472 33 528 97Z" id="g50-118"></path></g><g transform="matrix(.013,0,0,-0.013,16.22,-6.668)"><path d="M162 0V316H102V0H162ZM309 0V316H249V0H309Z" id="g119-35"></path></g><g transform="matrix(.013,0,0,-0.013,16.22,-3.066)"><path d="M162 0V316H102V0H162ZM309 0V316H249V0H309Z" id="g119-35"></path></g><g transform="matrix(.013,0,0,-0.013,16.22,.535)"><path d="M162 0V316H102V0H162ZM309 0V316H249V0H309Z" id="g119-35"></path></g><g transform="matrix(.013,0,0,-0.013,16.22,4.136)"><path d="M162 0V316H102V0H162ZM309 0V316H249V0H309Z" id="g119-35"></path></g><g transform="matrix(.0091,0,0,-0.0091,21.603,4.36)"><path d="M1000 227C1000 351 905 451 780 451C658 451 590 380 530 279C465 385 388 451 261 451C168 451 38 376 38 210C38 98 121 -12 261 -12C381 -12 448 59 508 160C573 53 652 -12 780 -12C879 -12 1000 64 1000 227ZM485 199C434 109 370 25 271 25C188 25 133 117 133 241C133 355 189 414 249 414C357 414 426 302 485 199ZM903 204C903 70 852 25 790 25C681 25 611 136 552 240C603 330 669 414 768 414C855 414 903 321 903 204Z" id="g50-30"></path></g></svg></span></td><td class="align_center">CPU Time</td></tr><tr><td class="thead-hr" colspan="4"><hr/></td></tr><tr><td class="align_left">Uniform Mesh</td><td class="align_center">404101</td><td class="align_center">1.35e-3</td><td class="align_center">6392.5s</td></tr><tr><td class="align_left">Adaptive Mesh</td><td class="align_center">147664</td><td class="align_center">1.32e-3</td><td class="align_center">2218.3s</td></tr><tr><td class="align_left">Adaptive Interface Mesh</td><td class="align_center">18249</td><td class="align_center">1.28e-3</td><td class="align_center">120.8s</td></tr><tr class="table-tr"><td colspan="4"><hr class="tbody-hr"/></td></tr></table></td></tr></table>

<div>Numerical results of Example <a href="https://static.hindawi.com/articles/mpe/volume-2018/3721258/figures/#ex1" target="_blank">2</a> with different mesh.</div>

Mathematical Problems in Engineering

tab1

Table 1

Table 1: A Numerical Method for Solving Elliptic Interface Problems Using Petrov-Galerkin Formulation with Adaptive Refinement 