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 $\Gamma =\partial\Omega$ is the boundary of $\Omega$ and 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 show that the unique solution of the problem \eqref{eq:problemU} is $$ \forall x,y\in[0,1]^2, \qquad u(x,y) = \cos(\pi x)\cos(\pi y). $$ In order to solve problem \eqref{eq:problemU} with the finite elements method, we write the weak formulation of the probleme \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. Our solution is composed by 3 different files. - LaplacianNeumann.geo : GMSH file, used to build the domain (the square). 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. - data.geo : this (auxiliary) file contains the index number associated with the geometry. It is used to ensure that GMSH and GetDP have the same numbering of the domains.
