FreeFem++/Example

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As an example for using FreeFem++, we are going to solve a Poisson equation. For a given function f(x, y), find a function u(x, y) satisfying


\begin{align}
-\Delta u(x,y)  & = & f(x,y) & \quad\forall & (x,y) & \in \Omega \\
u(x,y)          & = & 0      & \quad\forall & (x,y) & \,\text{on}\, \partial\Omega \\
\end{align}

Here \partial\Omega is the boundary of the bounded open set

\Omega \subset R^2 \text{and} \Delta u = \tfrac{\partial^2u}{\partial x^2} + \tfrac{\partial^2u}{\partial y^2}

The following is a FreeFem++ program which computes u when f(x, y) = xy and \Omega is the unit disk. The boundary C = \partial\Omega is

C = {(x, y)| x = \cos(t)\,, y = \sin(t), 0 \le t \le 2\Pi}

[leonhard@euler06 ~]$ ls -ltr test.edp 
-rw-r--r-- 1 sfux T0000 283 Dec  6 12:56 test.edp
[leonhard@euler06 ~]$ cat test.edp 
border C(t=0,2*pi){x=cos(t); y=sin(t);}
mesh Th = buildmesh (C(50));
fespace Vh(Th,P1);
Vh u,v;
func f= x*y;
real cpu=clock();
solve Poisson(u,v,solver=LU) = int2d(Th)(dx(u)*dx(v) + dy(u)*dy(v)) - int2d(Th)( f*v) + on(C,u=0) ;
plot(u);
cout << " CPU time = " << clock()-cpu << endl;