From a70d02814cb01ff15cfa3c6a3e0572742fc2c26f Mon Sep 17 00:00:00 2001 From: David Wells Date: Wed, 8 May 2019 09:20:47 -0400 Subject: [PATCH] step-29: wrap very long lines. No changes other than reflowing. --- examples/step-29/doc/intro.dox | 57 ++++++++++++++++++++++++++++------ 1 file changed, 47 insertions(+), 10 deletions(-) diff --git a/examples/step-29/doc/intro.dox b/examples/step-29/doc/intro.dox index 35944314af..7dcadffaba 100644 --- a/examples/step-29/doc/intro.dox +++ b/examples/step-29/doc/intro.dox @@ -96,13 +96,22 @@ boundary condition will yield partial reflections, i.e. only parts of the wave will pass through the boundary as if it wasn't here whereas the remaining fraction of the wave will be reflected back into the domain. -If we are willing to accept this as a sufficient approximation to an absorbing boundary we finally arrive at the following problem for $u$: +If we are willing to accept this as a sufficient approximation to an absorbing +boundary we finally arrive at the following problem for $u$: @f{eqnarray*} -\omega^2 u - c^2\Delta u &=& 0, \qquad x\in\Omega,\\ c (n\cdot\nabla u) + i\,\omega\,u &=&0, \qquad x\in\Gamma_2,\\ u &=& 1, \qquad x\in\Gamma_1. @f} -This is a Helmholtz equation (similar to the one in step-7, but this time with ''the bad sign'') with Dirichlet data on $\Gamma_1$ and mixed boundary conditions on $\Gamma_2$. Because of the condition on $\Gamma_2$, we cannot just treat the equations for real and imaginary parts of $u$ separately. What we can do however is to view the PDE for $u$ as a system of two PDEs for the real and imaginary parts of $u$, with the boundary condition on $\Gamma_2$ representing the coupling terms between the two components of the system. This works along the following lines: Let $v=\textrm{Re}\;u,\; w=\textrm{Im}\;u$, then in terms of $v$ and $w$ we have the following system: +This is a Helmholtz equation (similar to the one in step-7, but this time with +''the bad sign'') with Dirichlet data on $\Gamma_1$ and mixed boundary +conditions on $\Gamma_2$. Because of the condition on $\Gamma_2$, we cannot just +treat the equations for real and imaginary parts of $u$ separately. What we can +do however is to view the PDE for $u$ as a system of two PDEs for the real and +imaginary parts of $u$, with the boundary condition on $\Gamma_2$ representing +the coupling terms between the two components of the system. This works along +the following lines: Let $v=\textrm{Re}\;u,\; w=\textrm{Im}\;u$, then in terms +of $v$ and $w$ we have the following system: @f{eqnarray*} \left.\begin{array}{ccc} -\omega^2 v - c^2\Delta v &=& 0 \quad\\ @@ -120,13 +129,20 @@ This is a Helmholtz equation (similar to the one in step-7, but this time with ' \end{array}\right\} &\;& x\in\Gamma_1. @f} -For test functions $\phi,\psi$ with $\phi|_{\Gamma_1}=\psi|_{\Gamma_1}=0$, after the usual multiplication, integration over $\Omega$ and applying integration by parts, we get the weak formulation +For test functions $\phi,\psi$ with $\phi|_{\Gamma_1}=\psi|_{\Gamma_1}=0$, after +the usual multiplication, integration over $\Omega$ and applying integration by +parts, we get the weak formulation @f{eqnarray*} --\omega^2 \langle \phi, v \rangle_{\mathrm{L}^2(\Omega)} + c^2 \langle \nabla \phi, \nabla v \rangle_{\mathrm{L}^2(\Omega)} - c \omega \langle \phi, w \rangle_{\mathrm{L}^2(\Gamma_2)} &=& 0, \\ --\omega^2 \langle \psi, w \rangle_{\mathrm{L}^2(\Omega)} + c^2 \langle \nabla \psi, \nabla w \rangle_{\mathrm{L}^2(\Omega)} + c \omega \langle \psi, v \rangle_{\mathrm{L}^2(\Gamma_2)} &=& 0. +-\omega^2 \langle \phi, v \rangle_{\mathrm{L}^2(\Omega)} ++ c^2 \langle \nabla \phi, \nabla v \rangle_{\mathrm{L}^2(\Omega)} +- c \omega \langle \phi, w \rangle_{\mathrm{L}^2(\Gamma_2)} &=& 0, \\ +-\omega^2 \langle \psi, w \rangle_{\mathrm{L}^2(\Omega)} ++ c^2 \langle \nabla \psi, \nabla w \rangle_{\mathrm{L}^2(\Omega)} ++ c \omega \langle \psi, v \rangle_{\mathrm{L}^2(\Gamma_2)} &=& 0. @f} -We choose finite element spaces $V_h$ and $W_h$ with bases $\{\phi_j\}_{j=1}^n, \{\psi_j\}_{j=1}^n$ and look for approximate solutions +We choose finite element spaces $V_h$ and $W_h$ with bases $\{\phi_j\}_{j=1}^n, +\{\psi_j\}_{j=1}^n$ and look for approximate solutions @f[ v_h = \sum_{j=1}^n \alpha_j \phi_j, \;\; w_h = \sum_{j=1}^n \beta_j \psi_j. @f] @@ -134,8 +150,25 @@ Plugging into the variational form yields the equation system @f[ \renewcommand{\arraystretch}{2.0} \left.\begin{array}{ccc} -\sum_{j=1}^n \left(-\omega^2 \langle \phi_i, \phi_j \rangle_{\mathrm{L}^2(\Omega)} +c^2 \langle \nabla \phi_i, \nabla \phi_j \rangle_{\mathrm{L}^2(\Omega)}\right)\alpha_j - \left(c\omega \langle \phi_i,\psi_j\rangle_{\mathrm{L}^2(\Gamma_2)}\right)\beta_j &=& 0 \\ -\sum_{j=1}^n \left(-\omega^2 \langle \psi_i, \psi_j \rangle_{\mathrm{L}^2(\Omega)} +c^2 \langle \nabla \psi_i, \nabla \psi_j \rangle_{\mathrm{L}^2(\Omega)}\right)\beta_j + \left(c\omega \langle \psi_i,\phi_j\rangle_{\mathrm{L}^2(\Gamma_2)}\right)\alpha_j &=& 0 +\sum_{j=1}^n +\left( +-\omega^2 \langle \phi_i, \phi_j \rangle_{\mathrm{L}^2(\Omega)} ++ c^2 \langle \nabla \phi_i, \nabla \phi_j \rangle_{\mathrm{L}^2(\Omega)} +\right) +\alpha_j +- \left( +c\omega \langle \phi_i,\psi_j\rangle_{\mathrm{L}^2(\Gamma_2)}\right)\beta_j +&=& 0 \\ +\sum_{j=1}^n +\left( +-\omega^2 \langle \psi_i, \psi_j \rangle_{\mathrm{L}^2(\Omega)} ++ c^2 \langle \nabla \psi_i, \nabla \psi_j \rangle_{\mathrm{L}^2(\Omega)} +\right)\beta_j ++ \left( +c\omega \langle +\psi_i,\phi_j\rangle_{\mathrm{L}^2(\Gamma_2)} +\right)\alpha_j +&=& 0 \end{array}\right\}\;\;\forall\; i =1,\ldots,n. @f] In matrix notation: @@ -143,8 +176,12 @@ In matrix notation: \renewcommand{\arraystretch}{2.0} \left( \begin{array}{cc} --\omega^2 \langle \phi_i, \phi_j \rangle_{\mathrm{L}^2(\Omega)} + c^2 \langle \nabla \phi_i, \nabla \phi_j \rangle_{\mathrm{L}^2(\Omega)} & -c\omega \langle \phi_i,\psi_j\rangle_{\mathrm{L}^2(\Gamma_2)} \\ -c\omega \langle \psi_i,\phi_j\rangle_{\mathrm{L}^2(\Gamma_2)} & -\omega^2 \langle \psi_{i}, \psi_j \rangle_{\mathrm{L}^2(\Omega)} + c^2 \langle \nabla \psi_{i}, \nabla \psi_j \rangle_{\mathrm{L}^2(\Omega)} +-\omega^2 \langle \phi_i, \phi_j \rangle_{\mathrm{L}^2(\Omega)} ++ c^2 \langle \nabla \phi_i, \nabla \phi_j \rangle_{\mathrm{L}^2(\Omega)} +& -c\omega \langle \phi_i,\psi_j\rangle_{\mathrm{L}^2(\Gamma_2)} \\ +c\omega \langle \psi_i,\phi_j\rangle_{\mathrm{L}^2(\Gamma_2)} +& -\omega^2 \langle \psi_{i}, \psi_j \rangle_{\mathrm{L}^2(\Omega)} ++ c^2 \langle \nabla \psi_{i}, \nabla \psi_j \rangle_{\mathrm{L}^2(\Omega)} \end{array} \right) \left( -- 2.39.5