From 78faca2b7ad9a46842ff1016b6d989d58bfbdf6c Mon Sep 17 00:00:00 2001 From: wolf Date: Tue, 18 Jul 2000 14:32:01 +0000 Subject: [PATCH] Fix some minor things. git-svn-id: https://svn.dealii.org/trunk@3180 0785d39b-7218-0410-832d-ea1e28bc413d --- .../step-9.data/intro.tex | 36 ++++++++++--------- 1 file changed, 19 insertions(+), 17 deletions(-) diff --git a/deal.II/doc/tutorial/chapter-2.step-by-step/step-9.data/intro.tex b/deal.II/doc/tutorial/chapter-2.step-by-step/step-9.data/intro.tex index 533336f1e4..b5de177771 100644 --- a/deal.II/doc/tutorial/chapter-2.step-by-step/step-9.data/intro.tex +++ b/deal.II/doc/tutorial/chapter-2.step-by-step/step-9.data/intro.tex @@ -1,6 +1,6 @@ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% intro.html was generated from this file -%% with latex2html and some handwork +%% with latex2html and some hand work %% (copying out the relevant parts from the %% generated html file, replacing IMG= %% by the proper path) @@ -15,7 +15,7 @@ In this example, our aims are the following: \begin{itemize} - \item solve the advection equation $-\beta \cdot \nabla u = f$; + \item solve the advection equation $\beta \cdot \nabla u = f$; \item show how we can use multiple threads to get quicker to the desired results if we have a multi-processor machine; \item develop a simple refinement criterion. @@ -23,13 +23,15 @@ In this example, our aims are the following: While the second aim is difficult to describe in general terms without reference to the code, we will discuss the other two aims in the following. The use of multiple threads will then be detailed at the -relevant places within the program. +relevant places within the program. Furthermore, there exists a report on this +subject, which is also available online from the ``Documentation'' section of +the deal.II homepage. \paragraph{Discretizing the advection equation.} In the present example program, we shall numerically approximate the solution of the advection equation $$ - -\beta \cdot \nabla u = f, + \beta \cdot \nabla u = f, $$ where $\beta$ is a vector field that describes advection direction and speed (which may be dependent on the space variables), $f$ is a source @@ -57,7 +59,7 @@ $\vec x\in\partial\Omega$. This definition is quite intuitive, since as $\vec n$ points outward, the scalar product with $\beta$ can only be negative if the transport direction $\beta$ points inward, i.e. at the inflow boundary. The mathematical theory states that we must not -pose any boundary condition an the outflow part of the boundary. +pose any boundary condition on the outflow part of the boundary. As it is stated, the transport equation is not stably solvable using the standard finite element method, however. The problem is that @@ -86,9 +88,9 @@ $$ - (\beta\cdot \vec n u_h, v_h)_{\partial\Omega_-} = - (f, v_h + \delta \beta\cdot\nabla v_h). + (f, v_h + \delta \beta\cdot\nabla v_h)_\Omega - - (\beta\cdot \vec n g, v_h)_{\partial\Omega_-} + (\beta\cdot \vec n g, v_h)_{\partial\Omega_-}. $$ Note that we have included the inflow boundary values into the weak form, and that the respective terms to the left hand side operator are @@ -103,7 +105,7 @@ $$ (\beta\cdot \vec n \varphi_i, \varphi_j)_{\partial\Omega_-}, $$ with basis functions $\varphi_i,\varphi_j$. However, this is a -pitfall that happens to every numerical analysist at least once +pitfall that happens to every numerical analyst at least once (including the author): we have here expanded the solution $u_h = u_i \varphi_i$, but if we do so, we will have to solve the problem @@ -135,7 +137,7 @@ as system matrix. We will assemble this matrix in the program. There remains the solution of this linear system of equations. As the resulting matrix is no more symmetric positive definite, we can't -employ the usual CG method any more. Suitable for the solution is +employ the usual CG method any more. Suitable for the solution of systems as the one at hand is the BiCGStab (bi-conjugate gradients stabilized) method, which is also available in deal.II, so we will use it. @@ -155,9 +157,9 @@ the following domain and functions (in $d=2$ space dimensions): &=& \left\{ \begin{array}{ll} - \frac 1{10 s^d} & - \text{for $|\vec x-\vec x_0|