From 83c819ffe1cfb8afef0f5c9410ae4cd29b5b526c Mon Sep 17 00:00:00 2001 From: Timo Heister Date: Wed, 1 Jun 2016 12:10:57 +0100 Subject: [PATCH] typos --- examples/step-56/doc/intro.dox | 16 ++++++++-------- 1 file changed, 8 insertions(+), 8 deletions(-) diff --git a/examples/step-56/doc/intro.dox b/examples/step-56/doc/intro.dox index 1d2b8581f8..d3eb87877a 100644 --- a/examples/step-56/doc/intro.dox +++ b/examples/step-56/doc/intro.dox @@ -7,7 +7,7 @@ The purpose of this tutorial is to create an efficient linear solver for the Stokes equation and compare it to alternative -approaches. Using FGMRES with geometric multigrid as a precondtioner +approaches. Using FGMRES with geometric multigrid as a preconditioner for the velocity block, we see that the linear solvers used in step-22 cannot keep up since multigrid is the only way to get $O(n)$ solve time. Using the Timer class, we collect some statistics to compare @@ -25,7 +25,7 @@ and $p \in L_*^2 = \{ p \in L^2(\Omega), \int_\Omega p = 0 @f} Note that we are using the deformation tensor instead of $\Delta u$ (a -detailed desription of the difference between the two can be found in +detailed description of the difference between the two can be found in step-22, but in summary, the deformation tensor is more physical as well as more expensive). @@ -78,10 +78,10 @@ Since $P$ is aimed to be a preconditioner only, we shall use the approximations on the right in the equation above. As discussed in step-22, $-M_p^{-1}=\widetilde{S^{-1}} \approx -S^{-1}$, where $M_p$ is the pressure mass matrix and is solved approximatively by using CG +S^{-1}$, where $M_p$ is the pressure mass matrix and is solved approximately by using CG with ILU, and $\widetilde{A^{-1}}$ is obtained by one of multiple methods: CG with ILU as preconditioner, just using ILU, CG with GMG (Geometric -Multigrid as described in step-16) as a precondtioner, or just performing a single V-cycle +Multigrid as described in step-16) as a preconditioner, or just performing a single V-cycle of GMG. As a comparison, instead of FGMRES, we also use the direct solver @@ -92,8 +92,8 @@ the one dimensional null space given by the constant pressures, we fix the first

Reference Solution

-The testproblem is a "Manufactured Solution" (see step-7 for details). -We apply Dirichlet boundary condtions for the velocity on the whole +The test problem is a "Manufactured Solution" (see step-7 for details). +We apply Dirichlet boundary conditions for the velocity on the whole boundary of the domain $\Omega=[0,1]\times[0,1]\times[0,1]$. To enforce the boundary conditions we can just use our reference solution that we will now define. @@ -131,7 +131,7 @@ z) \sin(\pi x) \cos(\pi y)) @f} Because we do not enforce the mean pressure to be zero for our numerical solution in the linear system, -we need to postprocess the solution after solving. To do this we use +we need to post process the solution after solving. To do this we use the VectorTools::compute_mean_value() function to compute the mean value of the pressure to subtract it from the pressure. @@ -142,7 +142,7 @@ finite element system for the velocity. Since this is now part of the entire system, it is no longer easy to access. The reason for this is that there is currently no way in deal.II to ask, "May I have just part of a DoFHandler?" So in order to answer this request for our -needs, we have to create a new DoFHandler for just the velocites and +needs, we have to create a new DoFHandler for just the velocities and assure that it has the same ordering as the DoFHandler for the entire system so that we can copy over solution vectors element by element. -- 2.39.5