title = "Manufactured Solution for Laplace's Equation with Python"
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@@ -33,14 +33,14 @@ We solve Laplace's Equation in 2D on a $1 \times 1$ square domain.
Laplace's equation is
$$
\begin{equation}
-\mathop{\mathrm{div}} (a \mathop{\mathrm{grad}} u) = 0
\-\mathop{\mathrm{div}} (a \mathop{\mathrm{grad}} u) = 0
\end{equation}
$$
The weak form is derived as usual by multiplying with a test function $v$ and
integrating over the domain $\Omega$:
$$
\begin{equation}
-\int_{\Omega} v \mathop{\mathrm{div}} (a \mathop{\mathrm{grad}} u) \,\mathrm{d}\Omega = 0
\-\int_{\Omega} v \mathop{\mathrm{div}} (a \mathop{\mathrm{grad}} u) \,\mathrm{d}\Omega = 0
\,,
\end{equation}
$$
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@@ -50,13 +50,14 @@ $$
\int_{\Omega} a \mathop{\mathrm{grad}} v \cdot \mathop{\mathrm{grad}} u \,\mathrm{d}\Omega = \int_{\Omega} \mathop{\mathrm{div}} (v a \mathop{\mathrm{grad}} u) \,\mathrm{d}\Omega = \int_{\Gamma_{\mathrm{N}}} v a \mathop{\mathrm{grad}} u \cdot n \,\mathrm{d}\Gamma \,,
\end{equation}
$$
where in the second equality Gauss's theorem has been applied.
As usual, the domain boundary $\partial\Omega = \Gamma_{\mathrm{D}} \cup \Gamma_{\mathrm{N}}$ is subdivided
As usual, the domain boundary $\partial\Omega = \Gamma\_{\mathrm{D}} \cup \Gamma\_{\mathrm{N}}$ is subdivided
into the dirichlet and the Neumann boundary and $v$ vanishes on
$\Gamma_{\mathrm{D}}$.
$\Gamma\_{\mathrm{D}}$.
The r.h.s. of the above equation is the total flux associated with $u$ flowing
**into** the domain $\Omega$ through $\Gamma_{\mathrm{N}}$:
$-a \mathop{\mathrm{grad}} u$ is the flux density and $-n$ is the inwards directed surface
**into** the domain $\Omega$ through $\Gamma\_{\mathrm{N}}$:
$-a \mathop{\mathrm{grad}} u$ is the flux density and $\-n$ is the inwards directed surface
normal.
The weak form just derived is implemented (after FEM discretization) in the
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@@ -64,7 +65,7 @@ groundwater flow process in OpenGeoSys.
Note that for the application of Neumann boundary conditions, it is necessary to
know whether the flux has to be applied with a positive or a negative sign!
## Analytial solution
## Analytical solution
The coefficient $a$ of Laplace's equation is taken to be unity.
By differentiation it can be easily checked that
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@@ -77,7 +78,7 @@ solves Laplace's equation inside $\Omega$ for any $b$.
In this example we set $b = \tfrac 23 \pi$.
As boundary conditions we apply Dirichlet BCs at the top, left and bottom of the
domain with values from $u(x,y)|_{\Gamma_{\mathrm{D}}}$.
domain with values from $u(x,y)|\_{\Gamma_{\mathrm{D}}}$.
On the right boundary of the domain a Neumann BC is applied.
There $n = (1, 0)$, which implies that $a \mathop{\mathrm{grad}} u \cdot n
