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\section{Non-homogeneous Dirichlet conditions}
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\cindex{Poisson problem}
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\cindex{Dirichlet boundary condition}
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\cindex{Lagrange interpolation}
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\subsection*{Formulation}
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We turn now to the case of a non-homogeneous
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Dirichlet 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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$(P_2)_h$ {\it find $u$, defined in $\Omega$ such that:}
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-\Delta u &=& f \ {\rm in}\ \Omega \\
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u &=& g \ {\rm on}\ \partial \Omega
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The variationnal formulation of this problem expresses:
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$(VF_2)$ {\it find $u \in V$ such that:}
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a(u,v) = m(f,v), \ \forall v \in V_0
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a(u,v) &=& \int_\Omega \nabla u . \nabla v \, dx \\
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m(u,v) &=& \int_\Omega u v \, dx \\
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V &=& \{ v \in H^1(\Omega); \ v_{|\partial \Omega} = g \} \\
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The computation of the right-hand side may be considered with
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attention since $f$ is not {\em a priori} piecewise polynomial.
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\subsection*{Approximation}
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\cindex{approximation}
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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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V_h &=& \{ v \in X_h; \ v_{|\partial\Omega} = \pi_h(g) \} \\
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V_{0,h} &=& \{ v \in X_h; \ v_{|\partial\Omega} = 0 \} \\
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The approximate problem writes:
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{\it $(VF_2)_h$: find $u_h\in V_h$ such that:}
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a(u_h,v_h) = m(\Pi_h(f),v_h)
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\ \forall v_h \in V_{0,h}
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Notices that the evaluation of the right-hand side is performed by
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remplacing $f$ by its Lagrange interpolation $\Pi_h(h)$ on $\Omega$,
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while the boundary condition $g$ is remplaced by its Lagrange
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interpolation $\pi_h(h)$ on $\partial \Omega$.
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Let us choose $\Omega \subset R^N$, $N=1,2,3$ and
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f(x) &=& N \sin(\sum_{i=1}^N x_i) \\
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g(x) &=& \sin(\sum_{i=1}^N x_i)
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This choice is convenient, since
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the exact solution is known:
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$$u(x) = \sin(\sum_{i=1}^N x_i)$$.
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The following C++ code
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implement this problem in the \code{rheolef} environment.
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\subsection*{File \file{dirichlet-nh.cc}}
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\exindex{dirichlet-nh.cc}
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\verbatiminput{dirichlet-nh.cc}
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\subsection*{Comments}
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The code looks like the previous one, related to
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homogeneous boundary conditions.
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Let us comments the changes.
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The class \code{point} describes the coordinates
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of a point $(x_1, \ldots, x_N) \in R^N$ as a $N$-uplet
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The \code{Float} type is usually a \code{double}.
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This type depends upon the \code{rheolef}
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configuration \pxref{Installing,,Configure options},
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and could represent some high precision floating point
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class such as \code{doubledouble} (quadruple)
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or \code{bigfloat} (arbitrary).
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field fh = interpolate(Vh,f);
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Lagrange interpolation operator $\Pi_h (f)$.
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space Wh (omega, omega["boundary"], argv[2]);
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The space of picewise $P_k$ functions is builded.
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This space is suitable for the interpolation of $g$
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uh[boundary] = interpolate(Wh, g);
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The values of the degrees of freedom
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related to the boundary are stored into the field \code{u}, where
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non-homogeneous Dirichlet conditions applies.
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The rest of the code is similar to the homogeneous Dirichlet case.
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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 remplacing \code{dirichlet} by \code{dirichlet-nh}:
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mkgeo_grid -e 10 -boundary > line.geo
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./dirichlet-nh line.geo P1 | field -
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for the bidimensional case:
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mkgeo_grid -t 10 -boundary > square.geo
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./dirichlet-nh square.geo P1 | field -
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and for the tridimensional case:
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mkgeo_grid -T 10 -boundary > box.geo
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./dirichlet-nh box.geo P1 | field -mayavi -
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Here, the \code{P1} approximation can be remplaced by
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the \code{P2} one, by modifying the command-line argument.
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doc/usrman/dirichlet-nh.tex
