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\section{Non-homogeneous Neumann boundary conditions for the Helmholtz operator}
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\label{sec-neummann-nh}
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\cindex{operator!Helmholtz}%
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\subsection*{Formulation}
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Let us show how to insert Neumann boundary conditions.
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Let $f \in H^{-1}(\Omega)$ and
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$g \in H^{-\frac{1}{2}}(\partial \Omega)$.
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$(P_3)$: {\it find $u$, defined in $\Omega$ such that:}
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u -\Delta u &=& f \ {\rm in}\ \Omega \\
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\Frac{\partial u}{ \partial n} &=& g \ {\rm on}\ \partial \Omega
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The variational formulation of this problem expresses:
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$(VF_3)$: {\it find $u \in H^1(\Omega)$ such that:}
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a(u,v) = l(v), \ \forall v \in H^1(\Omega)
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a(u,v) &=& \int_\Omega u\,v \, {\rm d}x + \int_\Omega \nabla u . \nabla v \, {\rm d}x \\
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l(v) &=& \int_\Omega f\, v \, {\rm d}x + \int_{\partial \Omega} g \, v \, {\rm d}s
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\subsection*{Approximation}
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\cindex{Lagrange!interpolation}
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As usual, we introduce a mesh ${\cal T}_h$ of $\Omega$
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and the finite dimensional space $X_h$:
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X_h = \{ v \in H^1(\Omega); \
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\forall K \in {\cal T}_h \}
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The approximate problem writes:
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{\it $(VF_3)_h$: find $u_h\in X_h$ such that:}
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\subsection*{File \file{neumann-nh.cc}}
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\exindex{neumann-nh.cc}
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\myexample{neumann-nh.cc}
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Let us choose $\Omega \subset R^d$, $d=1,2,3$ and
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f(x) &=& (1+d\pi^2) \prod_{i=0}^{d-1} \sin(\pi x_i) \\
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-\pi & \mbox{ when } d = 1 \\
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-\pi\left( {\displaystyle \sum_{i=0}^{d-1} \sin(\pi x_i)}\right)
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& \mbox{ when } d = 2 \\
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-\pi\left( {\displaystyle \sum_{i=0}^{d-1} \sin(\pi x_i)} \sin(x_{(i+1){\rm mod}\,d}\right)
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This example is convenient, since
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the exact solution is known:
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u(x) = \prod_{i=0}^{d-1} \sin(\pi x_i)
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\subsection*{File \file{sinusprod\_helmholtz.icc}}
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\exindex{sinusprod\_helmholtz.icc}
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\myexample{sinusprod_helmholtz.icc}
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\subsection*{Comments}
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The {\tt neumann-nh.cc}
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code looks like the previous one {\tt dirichlet-nh.cc}.
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Let us comments only the changes.
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\begin{lstlisting}[numbers=none,frame=none]
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form m (Xh, Xh, "mass");
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form a (Xh, Xh, "grad_grad");
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The bilinear forms $m(.,.)$ and $a(.,.)$, also called the {\em mass} and
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{\em energy} in mechanics, are defined as:
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m(u,v) = \int_\Omega u\,v \, {\rm d}x
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a(u,v) = \int_\Omega \nabla u . \nabla v \, {\rm d}x
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Then, these two forms are added and stored in $a$:
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\begin{lstlisting}[numbers=none,frame=none]
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The right-hand side is computed as:
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\begin{lstlisting}[numbers=none,frame=none]
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field lh = riesz(Xh, f(d)) + riesz(Xh, "boundary", g(d));
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\subsection{How to run the program}
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First, compile the program:
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Running the program is obtained from the homogeneous Dirichlet case,
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by replacing \code{dirichlet} by \code{neumann-nh}.
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