<li> Initialization: Initial guess $u_0$ and $p_0$, tolerance $\tau$;
<li> Linear solve to compute update term $\delta\textbf{u}^{k}$ and $\delta p^k$;
<li> Update the approximation: $\textbf{u}^{k+1} = \textbf{u}^{k} + \delta\textbf{u}^{k}$ and $p^{k+1} = p^{k} + \delta p^{k}$;
- <li> Check residual norm: $E^{k+1} = \|F(\mathbf{u}^{k+1}, p^{k+1})\|$.
- If $E^{k+1} \leq \tau$, STOP.
- If $E^{k+1} > \tau$, back to step 2.
+ <li> Check residual norm: $E^{k+1} = \|F(\mathbf{u}^{k+1}, p^{k+1})\|$:
+ <ul>
+ <li>If $E^{k+1} \leq \tau$, STOP.
+ <li>If $E^{k+1} > \tau$, back to step 2.
+ </ul>
</ol>
<h3> Finding an Initial Guess </h3>
\right)
@f}
with a parameter $\gamma$ and an invertible matrix W. Here
-$\gamma B^TW^{-1}B & B^{T}$ is the Augmented Lagrangian term and
+$\gamma B^TW^{-1}B$ is the Augmented Lagrangian term and
see [1] for details.
Denoting the system matrix of the new system by $G$ and the right-hand
<th colspan="2">Mesh4</th>
</tr>
<tr>
- <th>Newton's iter </th>
+ <th>Newton iter </th>
<th>Residual </th>
<th>FGMRES </th>
<th>Residual </th>
<th>FGMRES </th>
</tr>
<tr>
- <th>1 </th>
- <th>7.40396e-3</th>
- <th>3 </th>
- <th>1.05562e-3 </th>
- <th>3 </th>
- <th>4.94796e-4 </th>
- <th>3 </th>
- <th>2.5624e-4 </th>
- <th>2 </th>
- <th>1.26733e-4 </th>
- <th>2 </th>
+ <td>1 </td>
+ <td>7.40396e-3</td>
+ <td>3 </td>
+ <td>1.05562e-3 </td>
+ <td>3 </td>
+ <td>4.94796e-4 </td>
+ <td>3 </td>
+ <td>2.5624e-4 </td>
+ <td>2 </td>
+ <td>1.26733e-4 </td>
+ <td>2 </td>
</tr>
<tr>
- <th>2 </th>
- <th>3.86766e-3 </th>
- <th>4 </th>
- <th>1.3549e-5 </th>
- <th>3 </th>
- <th>1.41981e-6 </th>
- <th>3 </th>
- <th>1.29108e-6 </th>
- <th>4 </th>
- <th>6.14794e-7 </th>
- <th>4 </th>
+ <td>2 </td>
+ <td>3.86766e-3 </td>
+ <td>4 </td>
+ <td>1.3549e-5 </td>
+ <td>3 </td>
+ <td>1.41981e-6 </td>
+ <td>3 </td>
+ <td>1.29108e-6 </td>
+ <td>4 </td>
+ <td>6.14794e-7 </td>
+ <td>4 </td>
</tr>
<tr>
- <th>3 </th>
- <th>1.60421e-3</th>
- <th>4 </th>
- <th>1.24836e-9 </th>
- <th>3 </th>
- <th>9.11557e-11 </th>
- <th>3 </th>
- <th>3.35933e-11 </th>
- <th>3 </th>
- <th>5.86734e-11 </th>
- <th>2 </th>
+ <td>3 </td>
+ <td>1.60421e-3</td>
+ <td>4 </td>
+ <td>1.24836e-9 </td>
+ <td>3 </td>
+ <td>9.11557e-11 </td>
+ <td>3 </td>
+ <td>3.35933e-11 </td>
+ <td>3 </td>
+ <td>5.86734e-11 </td>
+ <td>2 </td>
</tr>
<tr>
- <th>4 </th>
- <th>9.26748e-4 </th>
- <th>4 </th>
- <th>2.75537e-14 </th>
- <th>4 </th>
- <th>1.39986e-14 </th>
- <th>5 </th>
- <th>2.18864e-14 </th>
- <th>5 </th>
- <th>3.38787e-14 </th>
- <th>5 </th>
+ <td>4 </td>
+ <td>9.26748e-4 </td>
+ <td>4 </td>
+ <td>2.75537e-14 </td>
+ <td>4 </td>
+ <td>1.39986e-14 </td>
+ <td>5 </td>
+ <td>2.18864e-14 </td>
+ <td>5 </td>
+ <td>3.38787e-14 </td>
+ <td>5 </td>
</tr>
<tr>
- <th>5 </th>
- <th>1.34601e-5</th>
- <th>4 </th>
- <th> </th>
- <th> </th>
- <th> </th>
- <th> </th>
- <th> </th>
- <th> </th>
- <th> </th>
- <th> </th>
+ <td>5 </td>
+ <td>1.34601e-5</td>
+ <td>4 </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
</tr>
<tr>
- <th>6 </th>
- <th>2.5235e-8 </th>
- <th>5 </th>
- <th> </th>
- <th> </th>
- <th> </th>
- <th> </th>
- <th> </th>
- <th> </th>
- <th> </th>
- <th> </th>
+ <td>6 </td>
+ <td>2.5235e-8 </td>
+ <td>5 </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
</tr>
<tr>
- <th>7 </th>
- <th>1.38899e-12 </th>
- <th>4 </th>
- <th> </th>
- <th> </th>
- <th> </th>
- <th> </th>
- <th> </th>
- <th> </th>
- <th> </th>
- <th> </th>
+ <td>7 </td>
+ <td>1.38899e-12 </td>
+ <td>4 </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
</tr>
<tr>
- <th>8 </th>
- <th>4.68224e-15 </th>
- <th>4 </th>
- <th> </th>
- <th> </th>
- <th> </th>
- <th> </th>
- <th> </th>
- <th> </th>
- <th> </th>
- <th> </th>
+ <td>8 </td>
+ <td>4.68224e-15 </td>
+ <td>4 </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
</tr>
</table>
-The following figures show the sequence of the generated grids. For the case
+The following figures show the sequence of generated grids. For the case
of Re=400, the initial guess is obtained by solving Stokes on an $8 \times 8$
mesh, and the mesh is refined adaptively. Between meshes, the solution from
the coarse mesh is interpolated to the fine mesh to be used as an initial guess.
<img src="https://www.dealii.org/images/steps/developer/step-57.Re7500_get_convergence.svg" style="width:50%" alt="">
-Also we show the residual from each step of Newton's iteration on every
-mesh. The quadratic convergence is shown clearly in the table.
+We also show the residual from each step of Newton's iteration on every
+mesh. The quadratic convergence is clearly visible in the table.
<table align="center" border="1">
-
<tr>
<th> </th>
<th colspan="2">Mesh0</th>
<th colspan="2">Mesh3</th>
<th colspan="2">Mesh4</th>
</tr>
-
<tr>
- <th>Newton's iter </th>
+ <th>Newton iter </th>
<th>Residual </th>
<th>FGMRES </th>
<th>Residual </th>
<th>Residual </th>
<th>FGMRES </th>
</tr>
-
<tr>
- <th>1 </th>
- <th>1.89223e-6 </th>
- <th>6 </th>
- <th>4.2506e-3 </th>
- <th>3 </th>
- <th>1.42993e-3 </th>
- <th>3 </th>
- <th>4.87932e-4 </th>
- <th>2 </th>
- <th>1.89981e-04 </th>
- <th>2 </th>
+ <td>1 </td>
+ <td>1.89223e-6 </td>
+ <td>6 </td>
+ <td>4.2506e-3 </td>
+ <td>3 </td>
+ <td>1.42993e-3 </td>
+ <td>3 </td>
+ <td>4.87932e-4 </td>
+ <td>2 </td>
+ <td>1.89981e-04 </td>
+ <td>2 </td>
</tr>
-
<tr>
- <th>2 </th>
- <th>3.16439e-9</th>
- <th>8 </th>
- <th>1.3732e-3 </th>
- <th>7 </th>
- <th>4.15062e-4 </th>
- <th>7 </th>
- <th>9.11191e-5 </th>
- <th>8 </th>
- <th>1.35553e-5</th>
- <th>8 </th>
+ <td>2 </td>
+ <td>3.16439e-9</td>
+ <td>8 </td>
+ <td>1.3732e-3 </td>
+ <td>7 </td>
+ <td>4.15062e-4 </td>
+ <td>7 </td>
+ <td>9.11191e-5 </td>
+ <td>8 </td>
+ <td>1.35553e-5</td>
+ <td>8 </td>
</tr>
-
<tr>
- <th>3 </th>
- <th>1.7628e-14</th>
- <th>9 </th>
- <th>2.19455e-4 </th>
- <th>6 </th>
- <th>1.78805e-5 </th>
- <th>6 </th>
- <th>5.26782e-7 </th>
- <th>7 </th>
- <th>9.37391e-9 </th>
- <th>7 </th>
+ <td>3 </td>
+ <td>1.7628e-14</td>
+ <td>9 </td>
+ <td>2.19455e-4 </td>
+ <td>6 </td>
+ <td>1.78805e-5 </td>
+ <td>6 </td>
+ <td>5.26782e-7 </td>
+ <td>7 </td>
+ <td>9.37391e-9 </td>
+ <td>7 </td>
</tr>
-
<tr>
- <th>4 </th>
- <th> </th>
- <th> </th>
- <th>8.82693e-6 </th>
- <th>6 </th>
- <th>6.82096e-9 </th>
- <th>7 </th>
- <th>2.27696e-11 </th>
- <th>8 </th>
- <th>1.25899e-13</th>
- <th>9 </th>
+ <td>4 </td>
+ <td> </td>
+ <td> </td>
+ <td>8.82693e-6 </td>
+ <td>6 </td>
+ <td>6.82096e-9 </td>
+ <td>7 </td>
+ <td>2.27696e-11 </td>
+ <td>8 </td>
+ <td>1.25899e-13</td>
+ <td>9 </td>
</tr>
-
<tr>
- <th>5 </th>
- <th> </th>
- <th> </th>
- <th>1.29739e-7</th>
- <th>7 </th>
- <th>1.25167e-13 </th>
- <th>9 </th>
- <th>1.76128e-14 </th>
- <th>10 </th>
- <th> </th>
- <th> </th>
+ <td>5 </td>
+ <td> </td>
+ <td> </td>
+ <td>1.29739e-7</td>
+ <td>7 </td>
+ <td>1.25167e-13 </td>
+ <td>9 </td>
+ <td>1.76128e-14 </td>
+ <td>10 </td>
+ <td> </td>
+ <td> </td>
</tr>
-
<tr>
- <th>6 </th>
- <th> </th>
- <th> </th>
- <th>4.43518e-11</th>
- <th>7 </th>
- <th> </th>
- <th> </th>
- <th> </th>
- <th> </th>
- <th> </th>
- <th> </th>
+ <td>6 </td>
+ <td> </td>
+ <td> </td>
+ <td>4.43518e-11</td>
+ <td>7 </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
</tr>
-
<tr>
- <th>7 </th>
- <th> </th>
- <th> </th>
- <th>6.42323e-15 </th>
- <th>9 </th>
- <th> </th>
- <th> </th>
- <th> </th>
- <th> </th>
- <th> </th>
- <th> </th>
+ <td>7 </td>
+ <td> </td>
+ <td> </td>
+ <td>6.42323e-15 </td>
+ <td>9 </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
+ <td> </td>
</tr>
</table>
The sequence of generated grids looks like this:
<table align="center" border="1">
<tr>
- <td>Ref</td>
- <td>DoFs</td>
- <td>Iterative: Total/s (Setup/s)</td>
- <td>Direct: Total/s (Setup/s)</td>
+ <th>Ref</th>
+ <th>DoFs</th>
+ <th>Iterative: Total/s (Setup/s)</th>
+ <th>Direct: Total/s (Setup/s)</th>
</tr>
-
<tr>
<td>5</td>
<td>9539</td>
<td>0.10 (0.06)</td>
<td>0.13 (0.12)</td>
</tr>
-
<tr>
<td>6</td>
<td>37507</td>
<td>0.58 (0.37)</td>
<td>1.03 (0.97)</td>
</tr>
-
<tr>
<td>7</td>
<td>148739</td>
<td>3.59 (2.73)</td>
<td>7.78 (7.53)</td>
</tr>
-
<tr>
<td>8</td>
<td>592387</td>
<td>29.17 (24.94)</td>
<td>(>4GB RAM)</td>
</tr>
-
</table>
// Schur complement preconditioner is defined in this part. As discussed in
// the introduction, the preconditioner in Krylov iterative methods is
// implemented as a matrix-vector product operator. In practice, the Schur
- // complement preconditioner is decomposed as a product of three matrices(as
+ // complement preconditioner is decomposed as a product of three matrices (as
// presented in the first section). The $\tilde{A}^{-1}$ in the first factor
// involves a solve for the linear system $\tilde{A}x=b$. Here we solve
// this system via a direct solver for simplicity. The computation involved
{
pressure_mass_matrix.reinit(sparsity_pattern.block(1,1));
pressure_mass_matrix.copy_from(system_matrix.block(1,1));
+
+ // Note that settings this pressure block to zero is not identical to
+ // not assembling anything in this block, because this operation here
+ // will (incorrectly) delete diagonal entries that come in from
+ // hanging node constraints for pressure DoFs. This means that our
+ // whole system matrix will have rows that are completely
+ // zero. Luckily, FGMRES handles these rows without any problem.
system_matrix.block(1,1) = 0;
}
}
{
const ConstraintMatrix &constraints_used = initial_step ? nonzero_constraints : zero_constraints;
- SolverControl solver_control (system_matrix.m(),1e-4*system_rhs.l2_norm(), true);
+ SolverControl solver_control (system_matrix.m(), 1e-4*system_rhs.l2_norm(), true);
SolverFGMRES<BlockVector<double> > gmres(solver_control);
SparseILU<double> pmass_preconditioner;
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