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 $u(x,y) = g(x,y)$