From: hartmann Date: Wed, 15 Aug 2007 07:25:37 +0000 (+0000) Subject: Improved use of references in the text. X-Git-Url: https://gitweb.dealii.org/cgi-bin/gitweb.cgi?a=commitdiff_plain;h=6de85a5c3ec3ce88194c08664e0f215b6e7b43ab;p=dealii-svn.git Improved use of references in the text. git-svn-id: https://svn.dealii.org/trunk@14957 0785d39b-7218-0410-832d-ea1e28bc413d --- diff --git a/deal.II/examples/step-12/doc/intro.dox b/deal.II/examples/step-12/doc/intro.dox index 6a829a8487..d96538c885 100644 --- a/deal.II/examples/step-12/doc/intro.dox +++ b/deal.II/examples/step-12/doc/intro.dox @@ -94,14 +94,14 @@ consistent, i.e. and conservative, i.e. @f[ {\mathcal H}(v,w,{\bf n})=-{\mathcal H}(w,v,-{\bf n}). -\qquad\qquad\qquad\mathrm{[conservative]}@f] +\qquad\qquad\qquad\mathrm{[conservativity]}@f] This yields the following discontinuous Galerkin discretization: find $u_h\in V_h$ such that @f[ \sum_\kappa\left\{-({\mathcal F}(u_h),\nabla v_h)_\kappa+({\mathcal H}(u_h^+,u_h^-,{\bf n}), v_h)_{\partial\kappa}\right\}=(f,v_h)_\Omega, \quad\forall v_h\in V_h. \qquad\qquad\qquad\mathrm{[dg-general1]}@f] %Boundary conditions are realized by replacing $u_h^-$ on the inflow boundary $\Gamma_-$ by the boundary function $g$. -In the special case of the transport equation +In the special case of the [transport-equation] the numerical flux in its simplest form is given by @f[ @@ -114,7 +114,7 @@ is given by where on the inflow part of the cell the value is taken from the neighboring cell, $u_h^-$, and on the outflow part the value is taken from the current cell, $u_h^+$. Hence, the discontinuous Galerkin -scheme for the transport equation [transport-equation] is given +scheme for the [transport-equation] is given by: find $u_h\in V_h$ such that for all $v_h\in V_h$ following equation holds: @f[ @@ -162,7 +162,7 @@ other normal by $-{\bf n}$. This way we get $-{\bf n}$ in the second face term in [dg-general2] that finally produces the minus sign in the jump $[v_h]$ in equation [dg-general3]. -For the linear transport equation [transport-equation] +For the linear [transport-equation] equation [dg-general3] simplifies to @f[ -\sum_\kappa(u_h,{\mathbf \beta}\cdot\nabla v_h)_\kappa+\sum_e\left\{({\mathbf \beta}\cdot{\bf n}\, u_h, [v_h])_{e_+\setminus\Gamma}+({\mathbf \beta}\cdot{\bf n}\, u_h^-, [v_h])_{e_-\setminus\Gamma}\right\}=(f,v_h)_\Omega-({\mathbf \beta}\cdot{\bf n}\, g, v_h)_{\Gamma_-},