Laplace and Poisson equations

The Poisson equation in two dimensions is

\[ u_{xx} + u_{yy} = f(x,y), \] where \(\Delta u = f\) and \(\Delta\) is the Laplacian operator and sometimes called a forcing function. If \(f\) is 0, then it is the Laplace equation.

• archetype of an elliptic PDE
• no time appears
• often represent steady state
• must complement with boundary condition; only consider \(u(x,y) = g(x,y)\)