Tutorial/Laplace equation with Neumann boundary condition

Contents

The problem

We propose here to solve a first very simple academic example with GMSH and GetDP. The problem is the following: \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 $\Omega$ is the unit square, $\partial\Omega$ its boundary and $f$ the two-variables function defined by $$ \forall x,y\in [0,1]^2,\qquad f(x,y) = (1+2\pi^2)\cos(\pi x)\cos(\pi y) $$ Then, the unique solution of problem \eqref{eq:problemU} is $$ \forall x,y\in[0,1]^2, \qquad u(x,y) = \cos(\pi x)\cos(\pi y). $$ If $H^1(\Omega)$ denotes the usual Sobolev space, then the variationnal formulation of problem \eqref{eq:problemU} reads \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}

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.