https://cfd-online.com/W/index.php?title=Multigrid_methods&feed=atom&action=historyMultigrid methods - Revision history2024-03-29T13:35:21ZRevision history for this page on the wikiMediaWiki 1.16.5https://cfd-online.com/W/index.php?title=Multigrid_methods&diff=24971&oldid=prevJackbender: /* Introduction */2019-07-12T09:49:19Z<p><span class="autocomment">Introduction</span></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>==Introduction==</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>==Introduction==</div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>The convergence rate of standard iterative solvers (Gauss-<del class="diffchange diffchange-inline">Siedl</del>, Jacobi, SOR) has a tendency to 'stall', i.e. to fail in effectively reducing errors after a few number of iterations. The problem is more prominent when the meshes are refined. In fact, standard solvers behave much better on coarse grids! A close inspection of this behavior reveals that the convergence rate is a function of the error field frequency, i.e. the gradient of the error from node to node. If the error is distributed in a high frequency mode, the convergence rate is fast. However, after the first few iterations, the error field is smoothed out (low frequency) and the convergence rate deteriorates. <br></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>The convergence rate of standard iterative solvers (Gauss-<ins class="diffchange diffchange-inline">Seidel</ins>, Jacobi, SOR) has a tendency to 'stall', i.e. to fail in effectively reducing errors after a few number of iterations. The problem is more prominent when the meshes are refined. In fact, standard solvers behave much better on coarse grids! A close inspection of this behavior reveals that the convergence rate is a function of the error field frequency, i.e. the gradient of the error from node to node. If the error is distributed in a high frequency mode, the convergence rate is fast. However, after the first few iterations, the error field is smoothed out (low frequency) and the convergence rate deteriorates. <br></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>For many iterative methods, the number of iterations required to reach a converged solution is linearly proportional to number of nodes in one direction. This behavior could be rooted out to the fact that during the iteration process, the information travels only one grid size per iteration. While for proper convergence, the information has to travel back and forth several times. </div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>For many iterative methods, the number of iterations required to reach a converged solution is linearly proportional to number of nodes in one direction. This behavior could be rooted out to the fact that during the iteration process, the information travels only one grid size per iteration. While for proper convergence, the information has to travel back and forth several times. </div></td></tr>
</table>Jackbenderhttps://cfd-online.com/W/index.php?title=Multigrid_methods&diff=19620&oldid=prevPeter: Reverted edits by Bancream (talk) to last revision by Mmahendhran2013-07-05T08:46:43Z<p>Reverted edits by <a href="/Wiki/Special:Contributions/Bancream" title="Special:Contributions/Bancream">Bancream</a> (<a href="/W/index.php?title=User_talk:Bancream&action=edit&redlink=1" class="new" title="User talk:Bancream (page does not exist)">talk</a>) to last revision by <a href="/W/index.php?title=User:Mmahendhran&action=edit&redlink=1" class="new" title="User:Mmahendhran (page does not exist)">Mmahendhran</a></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>== External links ==</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>== External links ==</div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>* [http://www.mgnet.org MGnet] An excellent reference for multigrid methods. It contains tutorials, free codes, presentations, papers...<del class="diffchange diffchange-inline">[http://www.highereducation.co.th]</del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>* [http://www.mgnet.org MGnet] An excellent reference for multigrid methods. It contains tutorials, free codes, presentations, papers...</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>* {{Wikipedia article|page=Multigrid_method}}</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>* {{Wikipedia article|page=Multigrid_method}}</div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><i> Return to [[Numerical methods | Numerical Methods]] </i></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><i> Return to [[Numerical methods | Numerical Methods]] </i></div></td></tr>
</table>Peterhttps://cfd-online.com/W/index.php?title=Multigrid_methods&diff=19619&oldid=prevBancream: /* External links */2013-07-05T07:35:33Z<p><span class="autocomment">External links</span></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>== External links ==</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>== External links ==</div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>* [http://www.mgnet.org MGnet] An excellent reference for multigrid methods. It contains tutorials, free codes, presentations, papers...</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>* [http://www.mgnet.org MGnet] An excellent reference for multigrid methods. It contains tutorials, free codes, presentations, papers...<ins class="diffchange diffchange-inline">[http://www.highereducation.co.th]</ins></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>* {{Wikipedia article|page=Multigrid_method}}</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>* {{Wikipedia article|page=Multigrid_method}}</div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><i> Return to [[Numerical methods | Numerical Methods]] </i></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><i> Return to [[Numerical methods | Numerical Methods]] </i></div></td></tr>
</table>Bancreamhttps://cfd-online.com/W/index.php?title=Multigrid_methods&diff=16412&oldid=prevMmahendhran: /* Definitions */2013-01-03T11:23:25Z<p><span class="autocomment">Definitions</span></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>A '''Prolongation''' operation is the inverse of restriction and is defined as the interpolation method used to inject the residual from the coarse grid to the finer one.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>A '''Prolongation''' operation is the inverse of restriction and is defined as the interpolation method used to inject the residual from the coarse grid to the finer one.</div></td></tr>
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<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>A typical multigrid cycle starts at a finest level. The fine level solution is then transferred to next coarser level, (restriction). After some relaxation cycles on the coarse level, the solution is then restricted to next coarser level until the coarsest level is reached. The solution obtained at the coarsest level is <del class="diffchange diffchange-inline">than </del>interpolated back to the finer level, (prolongation). The solution from this finer level is interpolated to next finer level after some relaxation iterations, called post multigrid sweeps. The solution is prologated till the finest level is reached. The whole process is repeated until satisfactory convergence is reached.</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>A typical multigrid cycle starts at a finest level. The fine level solution is then transferred to next coarser level, (restriction). After some relaxation cycles on the coarse level, the solution is then restricted to next coarser level until the coarsest level is reached. The solution obtained at the coarsest level is <ins class="diffchange diffchange-inline">then </ins>interpolated back to the finer level, (prolongation). The solution from this finer level is interpolated to next finer level after some relaxation iterations, called post multigrid sweeps. The solution is prologated till the finest level is reached. The whole process is repeated until satisfactory convergence is reached.</div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Subsequent sections will explain how agglomeration, restriction, and prolongation work.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Subsequent sections will explain how agglomeration, restriction, and prolongation work.</div></td></tr>
</table>Mmahendhranhttps://cfd-online.com/W/index.php?title=Multigrid_methods&diff=13840&oldid=prevPeter: Reverted edits by Yan King (talk) to last revision by Peter2012-02-06T13:50:09Z<p>Reverted edits by <a href="/Wiki/Special:Contributions/Yan_King" title="Special:Contributions/Yan King">Yan King</a> (<a href="/W/index.php?title=User_talk:Yan_King&action=edit&redlink=1" class="new" title="User talk:Yan King (page does not exist)">talk</a>) to last revision by <a href="/Wiki/User:Peter" title="User:Peter">Peter</a></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><i> Return to [[Numerical methods | Numerical Methods]] </i></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><i> Return to [[Numerical methods | Numerical Methods]] </i></div></td></tr>
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</table>Peterhttps://cfd-online.com/W/index.php?title=Multigrid_methods&diff=13839&oldid=prevYan King at 11:05, 6 February 20122012-02-06T11:05:54Z<p></p>
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</table>Yan Kinghttps://cfd-online.com/W/index.php?title=Multigrid_methods&diff=13837&oldid=prevPeter at 09:21, 6 February 20122012-02-06T09:21:04Z<p></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>== Classification ==</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>== Classification ==</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Multigrid methods are classified into two branches<br></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Multigrid methods are classified into two branches<br></div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div># '''<del class="diffchange diffchange-inline">Full </del>Geometric <del class="diffchange diffchange-inline">Multigrid or </del>FAS''' <br></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div># '''<ins class="diffchange diffchange-inline">[[</ins>Geometric <ins class="diffchange diffchange-inline">multigrid - </ins>FAS<ins class="diffchange diffchange-inline">]]</ins>''' <br></div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div># '''Algebraic <del class="diffchange diffchange-inline">Multigrid or </del>AMG''' <br></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div># '''<ins class="diffchange diffchange-inline">[[</ins>Algebraic <ins class="diffchange diffchange-inline">multigrid - </ins>AMG<ins class="diffchange diffchange-inline">]]</ins>''' <br></div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>In the Geometric Multigrid, agglomeration of the nodes (cells, elements, or control volumes) takes place on the geometric level, and a set of new data structures representing the coarse grids need to be constructed for each level. This method is usually difficult to deal with when using the finite volume method since the FVM is based on a cell centered discretization which makes it difficult to define irregularly shaped control volumes.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>In the Geometric Multigrid, agglomeration of the nodes (cells, elements, or control volumes) takes place on the geometric level, and a set of new data structures representing the coarse grids need to be constructed for each level. This method is usually difficult to deal with when using the finite volume method since the FVM is based on a cell centered discretization which makes it difficult to define irregularly shaped control volumes.</div></td></tr>
</table>Peterhttps://cfd-online.com/W/index.php?title=Multigrid_methods&diff=12995&oldid=prevGaryC: /* Introduction */ Spellcheck. "Igredient" to "Ingredient"2011-06-16T02:23:39Z<p><span class="autocomment">Introduction: </span> Spellcheck. "Igredient" to "Ingredient"</p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>[[Image:Multigrid_low-high_frequency_error(tsaad).jpg| Low-High frequency error as seen on fine and coarse grids]]</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>[[Image:Multigrid_low-high_frequency_error(tsaad).jpg| Low-High frequency error as seen on fine and coarse grids]]</div></td></tr>
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<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>Multigrid methods effectively reduce the distribution of low frequency errors which makes them the ideal <del class="diffchange diffchange-inline">igredient </del>to be used with standard solvers.</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>Multigrid methods effectively reduce the distribution of low frequency errors which makes them the ideal <ins class="diffchange diffchange-inline">ingredient </ins>to be used with standard solvers.</div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Note: Multigrid is '''NOT''' a solver. It is a technique used in conjuction with a linear solver to yield a better covergence rate.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Note: Multigrid is '''NOT''' a solver. It is a technique used in conjuction with a linear solver to yield a better covergence rate.</div></td></tr>
</table>GaryChttps://cfd-online.com/W/index.php?title=Multigrid_methods&diff=9005&oldid=prevLenalg: /* Introduction */2008-03-15T04:42:17Z<p><span class="autocomment">Introduction</span></p>
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<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>Multigrid methods effectively reduce the distribution of low frequency errors which makes them the ideal igredient to be used with standard <del class="diffchange diffchange-inline">solver</del>.</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>Multigrid methods effectively reduce the distribution of low frequency errors which makes them the ideal igredient to be used with standard <ins class="diffchange diffchange-inline">solvers</ins>.</div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Note: Multigrid is '''NOT''' a solver. It is a technique used in conjuction with a linear solver to yield a better covergence rate.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Note: Multigrid is '''NOT''' a solver. It is a technique used in conjuction with a linear solver to yield a better covergence rate.</div></td></tr>
</table>Lenalghttps://cfd-online.com/W/index.php?title=Multigrid_methods&diff=9004&oldid=prevLenalg: /* Introduction */2008-03-15T04:39:53Z<p><span class="autocomment">Introduction</span></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>==Introduction==</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>==Introduction==</div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>The convergence rate of standard iterative solvers (Gauss-Siedl, Jacobi, SOR) has a tendency to 'stall', i.e. to fail in effectively reducing errors after a few number of iterations. The problem is more prominent when the meshes are refined. In fact, standard <del class="diffchange diffchange-inline">solver </del>behave much better on coarse grids! A close inspection of this behavior reveals that the convergence rate is a function of the error field frequency, i.e. the gradient of the error from node to node. If the error is distributed in a high frequency mode, the convergence rate is fast. However, after the first few iterations, the error field is smoothed out (low frequency) and the convergence rate deteriorates. <br></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>The convergence rate of standard iterative solvers (Gauss-Siedl, Jacobi, SOR) has a tendency to 'stall', i.e. to fail in effectively reducing errors after a few number of iterations. The problem is more prominent when the meshes are refined. In fact, standard <ins class="diffchange diffchange-inline">solvers </ins>behave much better on coarse grids! A close inspection of this behavior reveals that the convergence rate is a function of the error field frequency, i.e. the gradient of the error from node to node. If the error is distributed in a high frequency mode, the convergence rate is fast. However, after the first few iterations, the error field is smoothed out (low frequency) and the convergence rate deteriorates. <br></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>For many iterative methods, the number of iterations required to reach a converged solution is linearly proportional to number of nodes in one direction. This behavior could be rooted out to the fact that during the iteration process, the information travels only one grid size per iteration. While for proper convergence, the information has to travel back and forth several times. </div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>For many iterative methods, the number of iterations required to reach a converged solution is linearly proportional to number of nodes in one direction. This behavior could be rooted out to the fact that during the iteration process, the information travels only one grid size per iteration. While for proper convergence, the information has to travel back and forth several times. </div></td></tr>
</table>Lenalg