Tutorial/Laplace equation with Neumann boundary condition
Contents
The considered problem
We propose here to solve a first very simple academic example with GMSH and GetDP. We considered the unit square $\Omega$ with boundary $\Gamma$ and unit outward normal $\mathbf{n}$. 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 }\Gamma, \end{cases} \end{equation} where the function $f$ is defined by $$ \forall x,y\in [0,1]^2,\qquad f(x,y) = (1+2\pi^2)\cos(\pi x)\cos(\pi y). $$ One can easily shows that the unique solution of the problem \eqref{eq:problemU} is $$ \forall x,y\in[0,1]^2, \qquad u(x,y) = \cos(\pi x)\cos(\pi y). $$ To solve problem \eqref{eq:problemU} using a the finite elements method, we write the weak formulation of the probleme \eqref{eq:problemU}, which reads as \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 $v$ are called "test-functions".
Outline of the program
We give here a (very) detailled solution. We proposed to construct 3 different files. The two principles are : - LaplacianSquare.geo : GMSH file, used to build the domain (the square). The extension ".geo" is mainly used to design a GMSH file - LaplacianSquare.pro : GetDP file, contains the weak formulation \eqref{eq:WeakFormulation} of the problem \eqref{eq:problemU}. The extension ".pro" is associated with GetDP files.
The last file, "data.geo", contains the index number associated with the geometry. It can be usefull to be sure that GMSH and GetDP have the same numbering of the domains.
