From 01a82d9a38929b82a9b81e724ab42d565ce5c24e Mon Sep 17 00:00:00 2001 From: bangerth Date: Mon, 3 Jan 2011 00:58:22 +0000 Subject: [PATCH] Minor edits. git-svn-id: https://svn.dealii.org/trunk@23101 0785d39b-7218-0410-832d-ea1e28bc413d --- deal.II/examples/step-38/doc/intro.dox | 20 ++++++++++++++------ 1 file changed, 14 insertions(+), 6 deletions(-) diff --git a/deal.II/examples/step-38/doc/intro.dox b/deal.II/examples/step-38/doc/intro.dox index 1640a4aef1..18cd414ad1 100644 --- a/deal.II/examples/step-38/doc/intro.dox +++ b/deal.II/examples/step-38/doc/intro.dox @@ -142,8 +142,9 @@ We produce one test case for a 2d problem and another one for 3d: \left[\nabla - \mathbf n (\mathbf n \cdot \nabla)\right] \cdot \left[\nabla - \mathbf n (\mathbf n \cdot \nabla)\right] - u. + u, @f] + where, of course, $\nabla - \mathbf n (\mathbf n \cdot \nabla)=\nabla_\Gamma$. Since we are on the unit circle, $\mathbf n=\mathbf x$. Furthermore, $\nabla u = \left(\begin{array}{c}-2x_2 \\ -2x_1\end{array}\right)$. Consequently, we have the following identities: @@ -199,7 +200,9 @@ We produce one test case for a 2d problem and another one for 3d: = -2 \frac{d}{dt}(-sin^2 t + \cos^2 t) = -2 (-2 \sin t \cos t - 2 \cos t \sin t) \\ - &= 8 \sin t \cos t, + &= 8 \sin t \cos t + \\ + &= 8 x_1x_2, @f} which is of course the same result as we had above. @@ -293,11 +296,13 @@ We produce one test case for a 2d problem and another one for 3d: In the program, we will also compute the $H^1$ seminorm error of the solution. Since the solution function and its numerical approximation are only defined on the manifold, the obvious definition of this error functional is -$| e |_{H^1} = \left( \int_\Omega | \left[\mathbf n \otimes \mathbf +$| e |_{H^1} + = | \nabla_\Gamma e |_{L_2} + = \left( \int_\Omega | \left[\mathbf 1 - \mathbf n \otimes \mathbf n\right]\nabla (u-u_h) |^2 \right)^{1/2}$. This requires us to provide the -tangential gradient $\left[\mathbf n \otimes \mathbf +tangential gradient $\left[\mathbf 1 - \mathbf n \otimes \mathbf n\right]\nabla u$ to the function VectorTools::integrate_difference -(introduced first in step-7), which we +(first introduced in step-7), which we will do by implementing the function Solution::gradient in the program below. @@ -324,4 +329,7 @@ functions, and we will get a boundary description for half circles or half spheres that is pre-defined in the library. The rest of the program follows closely step-4 and, as far as computing the -error, step-7. +error, step-7. Some aspects of this program, in particular the use of two +template arguments on the classes Triangulation, DoFHandler, and similar, are +already described in detail in step-34; you may wish to read through this +tutorial program as well. -- 2.39.5