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\\
\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]
@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:
\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(
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