View source for Tutorial/Laplace equation with Neumann boundary condition

== The considered problem ==


We propose here to solve a first very simple academic problem with the help of GMSH and GetDP. We considered the unit square $\Omega = [0,1]\times[0,1]$ and we seek $u$, solution of the following problem
\begin{equation}
\begin{cases}\label{eq:problemU}
\Delta u + u = f & \text{in } \Omega,\\
\displaystyle{\frac{\partial u}{\partial \mathbf{n}} = 0} & \text{on }\partial\Omega,
\end{cases}
\end{equation}
where $\mathbf{n}$ is the unit outward normal of $\Omega$ and $\displaystyle{\Delta = \frac{\partial^2}{\partial x_1^2} + \frac{\partial^2}{\partial x_2^2} }$ is the Laplacian operator.
[[File:LaplacianNeumannOmega.png|right|200px|frameless|Domain Omega.|alt=Domain Omega.]]
To simplify, we suppose that the function $f$ is defined by
$$
\forall (x,y)\in [0,1]^2,\qquad f(x,y) = (1+2\pi^2)\cos(\pi x)\cos(\pi y).
$$
With this choice of $f$, one can easily show that the unique solution of the problem \eqref{eq:problemU} reads as
$$
\forall (x,y)\in[0,1]^2, \qquad u(x,y) = \cos(\pi x)\cos(\pi y).
$$
In order to solve problem \eqref{eq:problemU} with the finite elements method, we write the weak formulation of problem \eqref{eq:problemU}:
\begin{equation}\label{eq:WeakFormulation}
\left\{\begin{array}{l}
\text{Find } u\in H^1(\Omega) \text{ such that, }\\
\displaystyle{\forall v\in H^1(\Omega), \qquad \int_{\Omega} \nabla u\cdot\nabla v \;{\rm d}\Omega + \int_{\Omega}uv \;{\rm d}\Omega - \int_{\Omega}fv\;{\rm d}\Omega = 0},
\end{array}\right.
\end{equation}
where $H^1(\Omega)$ is the classical Sobolev space and the functions $v$ are the ''test functions''.

== Outline of the program ==

We give here a (very) detailled solution which is composed by 3 different files:
; param.geo 
: This (auxiliary) file contains the index number associated with the geometry. This ensures that GMSH and GetDP use the same numbering of the domains.
; LaplacianNeumann.geo  
: GMSH file, used to build the domain (the square $\Omega$). The extension ".geo" is mainly used to design a GMSH file
; LaplacianNeumann.pro  
: GetDP file, contains the weak formulation \eqref{eq:WeakFormulation} of the problem \eqref{eq:problemU}. The extension ".pro" is associated with GetDP files.

== param.geo: the auxiliary file ==

{{tutorial|getdp/LaplacianNeumann/GMSH_GETDP/param.geo}}

== LaplacianNeumann.geo: creation of the geometry with GMSH ==

{{tutorial|getdp/LaplacianNeumann/GMSH_GETDP/LaplacianNeumann.geo|height=22em}}

== LaplacianNeumann.pro: weak formulation ==

{{tutorial|getdp/LaplacianNeumann/GMSH_GETDP/LaplacianNeumann.pro|height=22em}}

== How to launch ==

All the files (.geo and .pro) must be located in the same directory. 

; Meshing the domain:
* Graphical way: launch GMSH and open "LaplacianNeumann.geo". Then chose the "mesh" menu and click on "2D". Finally, do not forget to save the mesh by clicking on the "Save" button
* With a terminal: go to the directory and then type:
::: gmsh -LaplacianNeumann.geo -algo iso -2
:: Note that here we chose the isobare algorithm ("iso"). One can also chose the following algorithms:
::: del2d
::: frontal

After the mesh is built, a file LaplacianNeumann.msh should have been created in the directory.

; Solving the problem with GetDP:
In a Terminal, type (in the right directory)
:getdp LaplacianNeumann.pro -solve -pos

GetDP will then propose the Resolution ("-solve") and the PostOperation ("-pos"). This should build a file called "u_Neumann.pos".

Remarks :
: the choice can be pre-selected by typing:
::getdp LaplacianNeumann.pro -solve#1 -pos#1
: There exists other options, like the choice of the linear solver, \ldots

; Showing the result

Finally, just launch GMSH and open the file "u_Neumann.pos" (graphical way or type "gmsh u_Neumann.pos" in a terminal in the right directory).

== Result ==

[[File:LaplacianNeumannmesh.jpg|400px|frameless|alt=Mesh of Omega.|Mesh of Omega.]]
[[File:LaplacianNeumannsolutionU.png|400px|frameless|alt=Modulus of u.|Modulus of u.]]

== All files ==

A zipfile containing all files (.geo, .pro, .mesh and .pos) can be downloaded [http://onelab.info/files/getdp/LaplacianNeumann/LaplacianNeumann.zip here].