FreeFem++/Example

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

$-\Delta u(x,y) = f(x,y) \,\forall\, (x,y) \in \Omega$

$u(x,y) = 0 \forall (x,y) \,\text{on}\, \partial\Omega$

Here ∂Ω is the boundary of the bounded open set

Ω ⊂ R^2 and ∆u = \tfrac{∂^2u}{∂x^2} + \tfrac{∂^2u∂y^2}

The following is a FreeFem++ program which computes u when f(x, y) = xy and Ω is the unit disk. The boundary C = ∂Ω is C = {(x, y)| x = cos(t), y = sin(t), 0 ≤ t ≤ 2π}

[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;