Difference between revisions of "Tutorial/Laplace equation with Neumann boundary condition"
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We propose here to solve a first very simple academic example with GMSH and GetDP. The problem is the following: | We propose here to solve a first very simple academic example with GMSH and GetDP. The problem is the following: | ||
\begin{equation} | \begin{equation} | ||
| − | \begin{cases} | + | \begin{cases}\label{eq:problemU} |
\Delta u + u = f & \text{in } \Omega\\ | \Delta u + u = f & \text{in } \Omega\\ | ||
\displaystyle{\frac{\partial u}{\partial \mathbf{n}} = 0} & \text{on }\partial\Omega, | \displaystyle{\frac{\partial u}{\partial \mathbf{n}} = 0} & \text{on }\partial\Omega, | ||
| Line 8: | Line 8: | ||
where $\Omega$ is the unit square, $\partial\Omega$ its boundary and $f$ the two-variables function defined by | where $\Omega$ is the unit square, $\partial\Omega$ its boundary and $f$ the two-variables function defined by | ||
$$ | $$ | ||
| − | f(x,y) = (1+2\pi^2)\cos(\pi x)\cos(\pi y) | + | \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). | ||
$$ | $$ | ||
Revision as of 10:56, 1 September 2011
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 $$ \forall x,y\in[0,1]^2, \qquad u(x,y) = \cos(\pi x)\cos(\pi y). $$
