From e80670b43844619494a0a23327e6bc531c3f853e Mon Sep 17 00:00:00 2001 From: hartmann Date: Wed, 1 Mar 2006 13:56:36 +0000 Subject: [PATCH] Fix two bugs. git-svn-id: https://svn.dealii.org/trunk@12520 0785d39b-7218-0410-832d-ea1e28bc413d --- .../chapter-2.step-by-step/step-12.data/intro.tex | 9 +++++++-- 1 file changed, 7 insertions(+), 2 deletions(-) diff --git a/deal.II/doc/tutorial/chapter-2.step-by-step/step-12.data/intro.tex b/deal.II/doc/tutorial/chapter-2.step-by-step/step-12.data/intro.tex index a465e22431..ab71ff1936 100644 --- a/deal.II/doc/tutorial/chapter-2.step-by-step/step-12.data/intro.tex +++ b/deal.II/doc/tutorial/chapter-2.step-by-step/step-12.data/intro.tex @@ -114,7 +114,10 @@ scheme for the transport equation \eqref{transport-equation} is given by: find $u_h\in V_h$ such that for all $v_h\in V_h$ following equation holds: \begin{equation}\label{dg-transport} - \sum_\kappa\left\{-(u_h,\boldsymbol\beta\cdot\nabla v_h)_\kappa+(\boldsymbol\beta\cdot{\bf n}\, u_h, v_h)_{\partial\kappa_+\setminus\Gamma}+(\boldsymbol\beta\cdot{\bf n}\, u_h^-, v_h)_{\partial\kappa_-\setminus\Gamma}\right\}=(f,v_h)_\Omega-(\boldsymbol\beta\cdot{\bf n}\, g, v_h)_{\Gamma_-}, + \sum_\kappa\left\{-(u_h,\boldsymbol\beta\cdot\nabla v_h)_\kappa + +(\boldsymbol\beta\cdot{\bf n}\, u_h, v_h)_{\partial\kappa_+} + +(\boldsymbol\beta\cdot{\bf n}\, u_h^-, v_h)_{\partial\kappa_-\setminus\Gamma}\right\} + =(f,v_h)_\Omega-(\boldsymbol\beta\cdot{\bf n}\, g, v_h)_{\Gamma_-}, \end{equation} where $\partial\kappa_-:=\{x\in\partial\kappa, \boldsymbol\beta(x)\cdot{\bf n}(x)<0\}$ denotes the inflow boundary @@ -123,7 +126,9 @@ outflow part of cell $\kappa$. Below, this equation will be referred to as \emph{first version} of the DG method. We note that after a second integration by parts, we obtain: find $u_h\in V_h$ such that \[ - \sum_\kappa\left\{(\nabla\cdot\{\boldsymbol\beta u_h\},\nabla v_h)_\kappa-(\boldsymbol\beta\cdot{\bf n} [u_h], v_h)_{\partial\kappa_-}\right\}=(f,v_h)_\Omega, \quad\forall v_h\in V_h, + \sum_\kappa\left\{(\nabla\cdot\{\boldsymbol\beta u_h\},v_h)_\kappa + -(\boldsymbol\beta\cdot{\bf n} [u_h], v_h)_{\partial\kappa_-}\right\} + =(f,v_h)_\Omega, \quad\forall v_h\in V_h, \] where $[u_h]=u_h^+-u_h^-$ denotes the jump of the discrete function between two neighboring cells and is defined to be $[u_h]=u_h^+-g$ on -- 2.39.5